System, apparatus and method for extracting three-dimensional information of an object from received electromagnetic radiation

ABSTRACT

An apparatus and method to produce a hologram of an object includes an electromagnetic radiation assembly configured to receive a received electromagnetic radiation, such as light, from the object. The electromagnetic radiation assembly is further configured to diffract the received electromagnetic radiation and transmit a diffracted electromagnetic radiation. An image capture assembly is configured to capture an image of the diffracted electromagnetic radiation and produce the hologram of the object from the captured image.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.13/970,103, filed Aug. 19, 2013, which is a continuation of U.S. patentapplication Ser. No. 12/515,343, filed Feb. 18, 2010 (now patent no.U.S. Pat. No. 8,542,421, issued Sep. 24, 2013), which is a nationalstage of PCT/US07/85094, filed Nov. 19, 2007, and claims the benefit ofpriority under 35 U.S.C. §119(e) of U.S. Patent Provisional ApplicationNo. 60/869,022, filed Dec. 7, 2006, and U.S. Patent ProvisionalApplication No. 60/866,358, filed Nov. 17, 2006. The contents of theseapplications are incorporated herein by reference in their entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to an apparatus for capturing electromagneticradiation, such as light or other forms of electromagnetic radiation,from an object and extracting object geometric information from thereceived radiation, in the field of three-dimensional imaging andholography. The invention also relates to a system and method ofperforming those functions.

2. Discussion of the Background

Conventional techniques for capturing three-dimensional information fromphysical objects include holography, range-finding, and tomography.However, conventional techniques may disadvantageously require an activeillumination source, or place limitations on a light source (e.g., mayrequire coherent light, a point light source or a bandwidth limitedlight), place limitations on movement of the object or the sensingapparatus (e.g., require that the object and sensing device bestationary, or require that they be moved in a predetermined fashion),may require complex electromagnetic radiation assemblies (e.g., complexarrangement of mirrors and lenses), and may produce poor qualitythree-dimensional images having low resolution or low fidelity.

SUMMARY OF THE INVENTION

Accordingly, one object of this invention is to provide an apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea received electromagnetic radiation from the object, diffract thereceived electromagnetic radiation, and transmit a diffractedelectromagnetic radiation; and an image capture assembly configured tocapture an image of the diffracted electromagnetic radiation, andproduce the hologram of the object from the captured image.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation includes light.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation apparatus includes only oneradiation propagation axis and is configured to propagateelectromagnetic radiation only along the radiation propagation axis inonly one direction.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation assembly includes pluralelectromagnetic radiation elements each having an axis of symmetryarranged along a same straight line.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation assembly includes pluralelectromagnetic radiation elements each having a geometric centerarranged along a same straight line.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation received from the object and theelectromagnetic radiation diffracted by the electromagnetic radiationassembly have a same radiation propagation axis.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation received from the object includesincoherent light.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation received from the object isproduced by the object.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation received from the object does notinterfere with an electromagnetic radiation that is not received fromthe object to produce the hologram of the object.

Another object of this invention is to provide a novel apparatus,wherein the object and the apparatus are configured to remain stationaryduring the capture of the image.

Another object of this invention is to provide a novel apparatus,wherein each portion of the electromagnetic apparatus is configured toremain stationary during the capture of the image.

Another object of this invention is to provide a novel apparatus,wherein at least one of the object or the apparatus is configured to bein motion during the capture of the image.

Another object of this invention is to provide a novel apparatus,wherein the hologram is produced from a single captured image.

Another object of this invention is to provide a novel apparatus,wherein the hologram is produced from plural captured images.

Another object of this invention is to provide a novel apparatus,wherein the hologram includes a Fresnel hologram.

Another object of this invention is to provide a novel apparatus,wherein the hologram includes an image hologram.

Another object of this invention is to provide a novel apparatus,wherein a phase and intensity of the diffracted electromagneticradiation is described by a convolution of the received electromagneticradiation and a Fresnel Zone Plate.

Another object of this invention is to provide a novel apparatus,wherein the hologram includes geometric information of the object, andthe geometric information includes, for each electromagnetic radiationradiating surface of the object, (i) a range distance between theelectromagnetic radiation radiating surface of the object and theelectromagnetic radiation assembly, (ii) a horizontal offset distance ofthe electromagnetic radiation radiating surface of the object, and (iii)a vertical offset distance of the electromagnetic radiation radiatingsurface of the object.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation assembly is configured to transmitthe electromagnetic radiation including a convolution of the receivedelectromagnetic radiation and a complex transmission function includinga linear summation of a first transformed pattern, a second transformedpattern and a third transformed pattern, the first transformed patternincluding a first shifted concentric ring pattern, the secondtransformed pattern including a second shifted concentric ring pattern,and the third transformed pattern including a third shifted concentricring pattern.

Another object of this invention is to provide a novel apparatus,wherein each of the first, second and third shifted concentric ringpatterns are shifted away from one another in a same plane of theelectromagnetic radiation assembly.

Another object of this invention is to provide a novel apparatus,wherein each of the first, second and third shifted concentric ringpatterns includes a Fresnel Zone Pattern or a portion of a Fresnel ZonePattern.

Another object of this invention is to provide a novel apparatus,wherein the portion of the Fresnel Zone Pattern includes a Fresnel ZonePattern having one or more rings removed, one or more extra rings added,one or more rings having a varied width, or one or more rings having aportion of the ring removed.

Another object of this invention is to provide a novel apparatus,wherein a phase of the Fresnel Zone Pattern or the portion of theFresnel Zone Pattern in each of the first, second and third shiftedconcentric ring pattern is different.

Another object of this invention is to provide a novel apparatus,wherein a predetermined thickness and coefficients of absorption orreflectance of the electromagnetic radiation assembly is configured tocontrol the phase and intensity of the diffracted light.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation assembly is configured to controlat least one of the phase or intensity of the transmittedelectromagnetic radiation by varying a thickness of a material throughwhich electromagnetic radiation passes.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation assembly further comprises: afirst electromagnetic radiation assembly configured to receive thereceived electromagnetic radiation from the object and transmit a firsttransformed electromagnetic radiation; a complex mask assemblyconfigured to receive the first transformed electromagnetic radiationfrom the first electromagnetic radiation assembly, and transmit acomplex masked electromagnetic radiation according to a complextransmission function; and a second electromagnetic radiation assemblyconfigured to receive the complex masked electromagnetic radiation fromthe mask assembly, and transmit a second transformed electromagneticradiation as the diffracted electromagnetic radiation.

Another object of this invention is to provide a novel apparatus,wherein the complex mask assembly further comprises: a mask controllerconfigured to vary the complex transmission function of theelectromagnetic radiation assembly over time, said mask controllerconfigured to vary the complex transmission function to be based on aFourier transform of a first Fresnel Zone Pattern at a first time, aFourier transform of a second Fresnel Zone Pattern at a second time, anda Fourier transform of a third Fresnel Zone Pattern at a third time.

Another object of this invention is to provide a novel apparatus,wherein the image capture assembly further comprises: a timingcontroller configured to capture a first partial image at the firsttime, a second partial image at the second time, and a third partialimage at the third time; and a summing unit configured to produce thehologram as a sum of the first partial image captured at the first time,the second partial image captured at the second time, and the thirdpartial image captured at the third time.

Another object of this invention is to provide a novel apparatus,further comprising: an electromagnetic radiation separating assemblyconfigured to separate the electromagnetic radiation received from theobject into three object electromagnetic radiation portions eachincluding a different frequency range; said first electromagneticradiation assembly including three first electromagnetic radiationsubassemblies each configured to receive one of the three objectelectromagnetic radiation portions, and respectively transmit first,second and third portions of the first transformed electromagneticradiation; said mask assembly including first, second and third masksubassemblies respectively configured to receive the first, second andthird portions of the first transformed electromagnetic radiation, andrespectively transmit first, second and third complex mask transformedelectromagnetic radiation; and said second electromagnetic radiationassembly including three second electromagnetic radiation subassembliesrespectively configured to receive first, second and third complex masktransformed electromagnetic radiation, and respectively transmit first,second and third portions of transmitted electromagnetic radiation.

Another object of this invention is to provide a novel apparatus,wherein the first mask subassembly is configured to transmit the firstcomplex mask transformed electromagnetic radiation based on a Fouriertransform of a first Fresnel Zone Pattern, the second mask subassemblyis configured to transmit the second complex mask transformedelectromagnetic radiation based on a Fourier transform of a secondFresnel Zone Pattern, and the third mask subassembly is configured totransmit the third complex mask transformed electromagnetic radiationbased on a Fourier transform of a third Fresnel Zone Pattern.

Another object of this invention is to provide a novel apparatus,wherein the image capture assembly includes at least one of a CCD, aCMOS light sensitive device, another electronic camera, a lightsensitive emulsion, or another photosensitive device.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation assembly consists of i) onediffractive electromagnetic radiation element and ii) one converginglens or mirror.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation assembly consists of i) onediffractive electromagnetic radiation element and ii) two converginglenses or two mirrors.

Another object of this invention is to provide a novel apparatus,further comprising: an objective assembly arranged between the objectand the electromagnetic radiation assembly and configured to collimate,focus, invert or modify the electromagnetic radiation from the object,prior to the received electromagnetic radiation being received at theelectromagnetic radiation assembly.

Another object of this invention is to provide a novel apparatus,wherein the objective assembly includes at least one of an objectivelens, a zoom lens, a macro lens, a microscope, a telescope, a prism, afilter, a monochromatic filter, a dichroic filter, a complex objectivelens, a wide-angle lens, a camera, a pin-hole, a light slit, or amirror.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation apparatus includes a diffractiveelectromagnetic radiation element or two lenses configured to produce anoff-axis Fresnel Zone pattern when the two lenses are illuminated by acoherent light.

Another object of this invention is to provide a novel apparatus,wherein the two lenses are arranged in a same plane perpendicular to anradiation propagation axis of the received electromagnetic radiation andthe two lenses have different focal lengths.

Another object of this invention is to provide a novel apparatus,wherein the two lenses are arranged in different planes perpendicular toan radiation propagation axis of the received electromagnetic radiation.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation includes at least one of an x-rayradiation, a microwave radiation, an infrared light, a radio frequencysignal or an ultraviolet light.

Another object of this invention is to provide a novel apparatus,wherein the electromagnetic radiation assembly and the image captureassembly do not include any reflective electromagnetic radiationelements.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea received electromagnetic radiation from the object along a radiationaxis in an electromagnetic radiation receiving direction, transmit atransmitted electromagnetic radiation along the radiation axis in theelectromagnetic radiation receiving direction, and interfere a firstportion of the transmitted electromagnetic radiation with a secondportion of the transmitted electromagnetic radiation the transmittedelectromagnetic radiation; and an image capture assembly configured tocapture an image of the transmitted electromagnetic radiationtransmitted along the optical axis in the electromagnetic radiationreceiving direction, and produce the hologram of the object from thecaptured image, wherein the radiation axis is a straight line.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea received electromagnetic radiation from the object and transmit atransmitted electromagnetic radiation based on the receivedelectromagnetic radiation, the transmitted electromagnetic radiationincluding the hologram of the object; and an image capture assemblyconfigured to capture an image of the transmitted electromagneticradiation and produce the hologram of the object from the capturedimage.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea received electromagnetic radiation from the object, transmit atransmitted electromagnetic radiation based on the receivedelectromagnetic radiation, and interfere a first portion of thetransmitted electromagnetic radiation with a second portion of thetransmitted electromagnetic radiation; and an opaque image captureassembly configured to capture an image of the transmittedelectromagnetic radiation produced by the interference of at least thefirst and second portions of the transmitted electromagnetic radiation,and produce the hologram of the object from the captured image, whereina center of the electromagnetic radiation assembly and a center of theimage capture assembly are arranged along a same straight line.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly consisting of onediffractive electromagnetic radiation element and configured to receivea received electromagnetic radiation from the object and transmit atransmitted electromagnetic radiation based on the receivedelectromagnetic radiation; and an image capture assembly configured tocapture an image of the transmitted electromagnetic radiation, andproduce the hologram of the object from the captured image.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea received electromagnetic radiation from the object, perform atransformation of the received electromagnetic radiation, and transmitthe transformed received electromagnetic radiation, the transformationincluding a convolution of a function representing an intensitydistribution of the received electromagnetic radiation and a concentricring function; and an image capture assembly configured to capture animage of the transmitted electromagnetic radiation, and produce thehologram of the object from the captured image.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea received electromagnetic radiation from the object, perform atransformation of the received electromagnetic radiation, and transmitthe transformed received electromagnetic radiation, the transformationincluding a convolution of i) an intensity distribution of the receivedelectromagnetic radiation and ii) a function having regions of positiveslope and negative slope when evaluated between a center of the opticalassembly and an outer edge of the optical assembly; and an image captureassembly configured to capture an image of the transmittedelectromagnetic radiation, and produce the hologram of the object fromthe captured image.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to convolvei) a received electromagnetic radiation received from the object and ii)a curve having plural inflection points between a center of the opticalassembly and an edge of the optical assembly, and transmit the convolvedelectromagnetic radiation; and an image capture assembly configured tocapture an image of the convolved electromagnetic radiation, and producethe hologram of the object from the captured image.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea received electromagnetic radiation from the object, perform atransformation of the received electromagnetic radiation, and transmitthe transformed received electromagnetic radiation, the transformationincluding a convolution of i) an intensity distribution of the receivedelectromagnetic radiation and ii) a transformation function that is alinear combination of three partial transformation functions, eachincluding a concentric ring pattern; and an image capture assemblyconfigured to capture an image of the transmitted electromagneticradiation, and produce the hologram of the object from the capturedimage.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of a chemiluminescent object, saidapparatus comprising: an electromagnetic radiation assembly configuredto receive a received chemiluminescent radiation from the object, andtransmit a transmitted electromagnetic radiation including the hologramof the object; and an image capture assembly configured to capture animage of the transmitted electromagnetic radiation, and produce thehologram of the object from the captured image.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea scattered electromagnetic radiation scattered by the object, whichscatters a source electromagnetic radiation, and transmit a transmittedelectromagnetic radiation based on the received scatteredelectromagnetic radiation, the transmitted electromagnetic radiationbeing independent of any source electromagnetic radiation that is notscattered by the object; and an image capture assembly configured tocapture an image of the transmitted electromagnetic radiation andproduce the hologram of the object from the captured image.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to diffractan electromagnetic radiation received from the object; and an imagecapture assembly configured to capture an image of the diffractedelectromagnetic radiation and produce the hologram of the object fromthe captured image.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: plural electromagnetic radiation sources configured toradiate the object with plural electromagnetic radiation signals; anelectromagnetic radiation assembly configured to receive a receivedelectromagnetic radiation from the object and transform the receivedelectromagnetic radiation, the received electromagnetic radiationincluding portions of the plural source electromagnetic radiationsignals scattered by the object; and a capture assembly configured tocapture an image of the transformed electromagnetic radiation andproduce the hologram of the object from the captured image.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of a fluorescent object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea received fluorescent radiation from the object and transmit atransmitted electromagnetic radiation based on the received fluorescentradiation; and an image capture assembly configured to capture an imageof the transmitted electromagnetic radiation and produce the hologram ofthe object from the captured image.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of a black body radiation radiatingobject, said apparatus comprising: an electromagnetic radiation assemblyconfigured to receive a received black body electromagnetic radiationfrom the object, and transmit a transmitted electromagnetic radiationbased on the received black body electromagnetic radiation from theobject; and an image capture assembly configured to capture an image ofthe transmitted electromagnetic radiation and produce the hologram ofthe object from the captured image.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea received electromagnetic radiation from the object, transmit atransmitted electromagnetic radiation based only on the receivedelectromagnetic radiation from the object, and interfere a first portionof the transmitted electromagnetic radiation with a second portion ofthe transmitted electromagnetic radiation; and an image capture assemblyconfigured to capture a fringe pattern produced by the interference ofat least the first and second portions of the transmittedelectromagnetic radiation and produce the hologram of the object fromthe fringe pattern.

Another object of this invention is to provide a novel electromagneticradiation apparatus configured to produce a hologram of an object, saidapparatus configured to receive a received electromagnetic radiationfrom the object, diffract the received electromagnetic radiation, andtransmit a diffracted electromagnetic radiation including the hologramof the object.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of a scene, said apparatus comprising:an electromagnetic radiation assembly configured to receive a receivedelectromagnetic radiation from the scene, diffract the receivedelectromagnetic radiation, and transmit a diffracted electromagneticradiation; and an image capture assembly configured to capture an imageof the diffracted electromagnetic radiation, and produce the hologram ofthe scene from the captured image.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea received electromagnetic radiation from the object, transform thereceived electromagnetic radiation, and transmit the transformedelectromagnetic radiation including a fringe pattern; and an imagecapture assembly configured to capture an image of the fringe patternand produce the hologram of the scene from the captured fringe pattern.

Another object of this invention is to provide a novel apparatusconfigured to produce a hologram of an object, said apparatuscomprising: an electromagnetic radiation assembly configured to receivea received electromagnetic radiation from the object and transform thereceived electromagnetic radiation; and an image capture assemblyconfigured to capture the transformed electromagnetic radiationincluding the hologram of the object, said hologram includes fringepatterns produced by an interference of the received electromagneticradiation with itself, and said hologram not including fringe patternsproduced by an interference of the received electromagnetic radiationwith any other electromagnetic radiation.

Another object of this invention is to provide a novel method forproducing a hologram of an object, said method comprising steps of:receiving a received electromagnetic radiation from the object;transmitting a diffracted electromagnetic radiation based on thereceived electromagnetic radiation; capturing an image of the diffractedelectromagnetic radiation; and producing the hologram of the object fromthe captured image.

Another object of this invention is to provide a novel apparatus,wherein the received electromagnetic radiation does not include coherentlight.

A new method of recording digital holograms under incoherentillumination reflects light from a three-dimensional (3-D) object,propagates through a diffractive optical element (DOE) and is recordedby a digital camera. Three holograms are recorded sequentially each fora different phase factor of the DOE. The three holograms are superposedin the computer such that the result is a complex valued Fresnelhologram. When this hologram is reconstructed in the computer, the 3-Dproperties of the object are revealed.

Another new imaging method records multicolor digital holograms fromobjects emitting fluorescent light. The fluorescent light specific tothe emission wavelength of various fluorescent dyes after excitation ofthree dimensional (3-D) objects is recorded on a digital monochromecamera after reflection from a diffractive optical element (DOE). Foreach wavelength of fluorescent emission, the camera sequentially recordsthree holograms reflected from the DOE, each with a different phasefactor of the DOE's function. The three holograms are superposed in acomputer to create a complex valued Fresnel hologram of each fluorescentemission. The holograms for each fluorescent color are further combinedin a computer to produce a multicolored fluorescence hologram and 3-Dcolor image.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the invention and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1 is a block diagram of an optical apparatus according to anembodiment of the present invention;

FIG. 2 is a block diagram that illustrates an example of capturedgeometric information according to embodiments of the present invention;

FIG. 3 is a block diagram of an incoherent correlator that may be usedas the optical assembly in the optical apparatus of FIG. 1;

FIG. 4 is a block diagram of an embodiment of an optical apparatus thatincludes an optical assembly;

FIG. 5A is a detailed front view of an embodiment of a mask thatincludes a DOE having an array of plural transform regions;

FIG. 5B is a cross-section view of a symmetrically arranged volumemodulated diffractive optical element structure;

FIG. 5C is a cross-section view of a volume modulated diffractiveoptical element structure;

FIG. 5D is a cross-section view of an index modulated diffractiveoptical element structure;

FIG. 5E is a cross-section view of a mixed mode diffractive opticalelement structure;

FIG. 5F is a cross-section view of a reflective volume modulateddiffractive optical element;

FIG. 5G is a cross-section view of a reflective volume modulateddiffractive optical element;

FIG. 6A is a detailed front view of another embodiment of a mask thatincludes a DOE having an array of plural transform regions;

FIG. 6B is a cross-section view of a mask including a transmission layerhaving varied transmissivity for regions adjacent to correspondingtransform regions;

FIG. 6C is a cross-section view of a mask having transform regionsconfigured to vary an amplitude of the received light;

FIG. 6D is a cross-section view of a mask having a printed overlay;

FIG. 6E is another embodiment of a mask configured to vary an amplitudeof the received light;

FIG. 6F is a block diagram of an embodiment of a mask including SLMs;

FIG. 6G is a block diagram of another embodiment of a mask includingSLMs;

FIG. 7 is an example of a binary Fresnel Zone Pattern;

FIG. 8A is an example of a sinusoidal FZP;

FIG. 8B is an example of another sinusoidal FZP;

FIG. 8C is an example of another sinusoidal FZP;

FIG. 9A is an example of a Fourier Transformed FZP pattern;

FIG. 9B is another example of a Fourier Transformed FZP pattern;

FIG. 9C is another example of a Fourier Transformed FZP pattern;

FIG. 10A is the amplitude portion of a complex transmission functionthat is a Fourier Transform of a linear combination of three maskfunctions;

FIG. 10B is the phase portion of a complex transmission function that isa Fourier Transform of a linear combination of mask functions;

FIG. 10C is an example of a pattern on a CCD when a point object ispresent at the input;

FIG. 11A is a block diagram of an embodiment of an image captureassembly;

FIG. 11B is a block diagram of another embodiment of an image captureassembly;

FIG. 12A is a view of an embodiment of a light intensity capture devicethat includes a charge coupled device having three distinct regions;

FIG. 12B is an example of a two-dimensional intensity image includingthree partial images;

FIG. 13A is an example of an arrangement of distinct regions in anembodiment of a light capturing device;

FIG. 13B is another example of an arrangement of distinct regions in anembodiment of a light capturing device;

FIG. 13C is another example of an arrangement of distinct regions in anembodiment of a light capturing device;

FIG. 14 is a block diagram of an embodiment of a capture control unitthat includes an image data processor that combines the electronic imagedata;

FIG. 15 is a block diagram of an embodiment of an optical apparatus thatvaries the mask over time;

FIG. 16 is a block diagram of a controllable mask that includes aspatial light modulator under the control of a mask controller;

FIG. 17 is a block diagram of another embodiment of an optical apparatusin which the mask is varied over time;

FIG. 18 is a block diagram of another embodiment of an opticalapparatus;

FIG. 19 is a block diagram of another embodiment of an opticalapparatus;

FIG. 20A is a block diagram of an embodiment of an optical apparatushaving a first transforming optical assembly including a reflectiveoptical assembly;

FIG. 20B is a block diagram of another embodiment of an opticalapparatus having a first transforming optical assembly including areflective optical assembly;

FIG. 21A is a block diagram of another embodiment of an opticalapparatus;

FIG. 21B is a block diagram of another embodiment of an opticalapparatus;

FIG. 22A is a block diagram of an example of an optical apparatus thatdoes not require a second transforming optical element;

FIG. 22B is a block diagram of an example of an optical apparatus thatdoes not require a first transforming optical element;

FIG. 22C is a block diagram of an example of an optical apparatus thatdoes not require first and second transforming optical elements;

FIG. 23 is a block diagram of an embodiment of an optical apparatusincluding a reflective type diffractive optical element;

FIG. 24A is a block diagram of another embodiment of an opticalapparatus;

FIG. 24B is a block diagram of another embodiment of an opticalapparatus;

FIG. 25 is a block diagram of another embodiment of an opticalapparatus;

FIG. 26 is an example of an off-axis Fresnel Zone Pattern;

FIG. 27 is a block diagram of a portion of an optical apparatusincluding a composite mask having lenses;

FIG. 28A is a side view of an embodiment of a composite mask;

FIG. 28B is a side view of another embodiment of a composite mask;

FIG. 28C is a side view of another embodiment of a composite mask;

FIG. 28D is a side view of another embodiment of a composite mask;

FIG. 29 is a block diagram of another embodiment of an opticalapparatus;

FIG. 30 is a detailed view of an embodiment of a grating having low andhigh transmissivity regions;

FIG. 31A is a block diagram of an embodiment of an optical apparatusthat may be used with a lined transparency or grating;

FIG. 31B is a block diagram of another embodiment of an opticalapparatus;

FIG. 32 is a block diagram of a conventional holographic system;

FIG. 33 is a block diagram of another embodiment of an opticalapparatus;

FIG. 34A shows a phase distribution of the reflection masks displayed onthe SLM with θ=0°;

FIG. 34B shows a phase distribution of the reflection masks displayed onthe SLM with θ=120°;

FIG. 34C shows a phase distribution of the reflection masks displayed onthe SLM with θ=240°;

FIG. 34D shows an enlarged portion of the reflection mask in FIG. 34Aindicating that half of the SLM's pixels (randomly chosen) modulatelight with constant phase;

FIG. 34E shows an enlarged portion of the reflection mask in FIG. 34Aindicating that half of the SLM's pixels (randomly chosen) modulatelight with constant magnitude;

FIG. 34F shows an enlarged portion of the reflection mask in FIG. 34Aindicating that half of the SLM's pixels (randomly chosen) modulatelight with phase of the final on-axis digital hologram;

FIG. 34G shows a reconstruction of the hologram of the three letters atthe best focus distance of ‘O’;

FIG. 34H shows a reconstruction of the hologram of the three letters atthe best focus distance of ‘S’;

FIG. 34I shows a reconstruction of the hologram of the three letters atthe best focus distance of ‘A’;

FIG. 35 is a block diagram of another embodiment of an opticalapparatus;

FIG. 36A shows magnitude of a complex Fresnel hologram of the dice;

FIG. 36B shows phase of a complex Fresnel hologram of the dice;

FIG. 36C shows a digital reconstruction of a non-fluorescence hologramat the face of the red-dots on the die;

FIG. 36D shows a digital reconstruction of a non-fluorescence hologramat the face of the green dots on the die;

FIG. 36E shows magnitude of a complex Fresnel hologram of the red dots;

FIG. 36F shows phase of a complex Fresnel hologram of the red dots;

FIG. 36G shows a digital reconstruction of the red fluorescence hologramat the face of the red-dots on the die;

FIG. 36H shows a digital reconstruction of the red fluorescence hologramat the face of the green dots on the die;

FIG. 36I shows magnitude of a complex Fresnel hologram of the greendots;

FIG. 36J shows phase of the complex Fresnel hologram of the green dots;

FIG. 36K shows a digital reconstruction of a green fluorescence hologramat the face of the red-dots on the die;

FIG. 36L shows a digital reconstruction of a green fluorescence hologramat the face of the green dots on the die;

FIG. 36M shows a composition of the digital reconstructions in FIGS.36C, 36G, and 36K; and

FIG. 36N shows a composition of the digital reconstructions in FIGS.36D, 36H, and 36L.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Conventional holographic techniques may include methods of capturing ahologram of an object by capturing an interference pattern that resultswhen a first portion of a coherent laser light beam (e.g., referencebeam) interferes with a second portion of the laser light beam reflectedoff the object (e.g., object beam). A three-dimensional image of theobject may be viewed by appropriately illuminating the recordedinterference pattern with the reference beam.

FIG. 32 is a block diagram of a conventional holographic systemincluding a laser 9000 that shines a coherent laser light beam along afirst optical axis 9026 through a partially reflective and transmissivemirror, such as a beam splitter 9002. A first portion of the split laserbeam is guided by lens 9004 and mirror 9008 to illuminate the object9014 with the object beam 9010 along a second optical axis 9024. Asecond portion of the split laser beam is reflected by the beamsplitter9002 along a third optical axis 9018 and guided by lens 9006 and mirror9028 to direct a reference beam 9012 along a fourth optical axis 9020 toan image capture device 9016 such as a photographic plate, a chargecoupled device (CCD) or a complementary metal oxide semiconductor sensor(CMOS). The reference beam 9012 and the object beam 9010 reflected fromthe object 9014 along a fifth optical axis 9022 interfere with eachother, producing an interference pattern that may be recorded as ahologram on the image capture device 9016.

Conventional holography solutions that produce a hologram by interferingtwo parts of a light source along different optical paths or opticalaxes may be very sensitive to any change in alignment of the opticalpaths or axes, because even minor changes in the length or direction ofthe optical paths or axes will change the phase relationship of theportions of light that interfere. Such a change will result in a changeto the resulting interference pattern and hologram, and yield adistorted resulting image.

For example, in the holography system of FIG. 32, a factor such asmotion, vibration, component deterioration or distortion, or thermalexpansion, may cause a slight change in the length or direction of oneor more of the first, second, third, fourth or fifth optical axes 9026,9024, 9018, 9020 and 9022, respectively. Even a slight change in one ofthe axes may cause a corresponding change in the phase relationship ofthe reference beam 9012 and the light reflected from the object 9014along the fifth optical axis 9010, thereby causing a significant changein the resulting interference pattern and hologram captured at the imagecapture device 9016.

Such a sensitivity to axial variation in conventional holographicsystems may result in reduced resolution in the resultingthree-dimensional information.

Various conventional attempts to address such a sensitivity to axialvariation have had limited success. For example, attempts have includedusing massive platforms and shock absorbers to dampen vibration, hightolerance mechanical optical stages to reduce positioning errors, andoptical and structural materials having reduced coefficients of thermalexpansion to reduce thermal expansion effects. However, these attemptsgenerally increase the cost, size and mass of conventional holographysystems, and reduce system portability and availability.

In addition, conventional holography systems may require an active lightsource to illuminate the object and produce the reference and objectbeams. Active solutions that require illumination of the object by aparticular light source may limit the applicability and usefulness ofthe conventional holography systems. For example, an active light sourcewould not be useful in stealth military targeting holographic systemswhere it would be undesirable for the targeting device to give away itsposition by producing light or other electromagnetic radiation.Alternatively, an active radiation source would not be applicable inholographic systems that observe objects that produce their own light,such as a holographic system observing chemiluminescent, black body, orinfrared illuminating objects. Such a holographic technique may beuseful in observing objects such as ships by virtue of their ability toblock the chemiluminescence of background emissions in certain bodies ofwater, such as the chemiluminescent Red Sea.

In addition, conventional holographic systems that rely on a coherentlight source, such as a monochromatic laser, may be unable to capturecolor information from the object unless multiline lasers or multiplelasers of different wavelengths are used. Systems such as those arelikely to be very complex. Further, such systems may not be suitable forcapturing three-dimensional information regarding objects that shouldnot be illuminated with laser light (e.g., sensitive biologicalmaterial).

Conventional holographic techniques using incoherent light to illuminatean object rely on a simplifying assumption that incoherent sourceobjects may be considered to be composed of many individual light sourcepoints, each of which is self coherent, and each of which can thereforecreate an interference pattern with its mirrored image. For the purposesof this document, incoherent light is any temporally or spatiallyincoherent light for which any two electromagnetic fields emitted from asame location at two different times (in the case of temporalincoherence) or emitted from two different locations at the same time(in the case of spatial incoherence) do not create an interferencegrating or pattern when the two fields are summed together. Variousmethods of incoherent holography have been proposed using thisprinciple, such as methods described in A. W. Lohmann, “WavefrontReconstruction for Incoherent Objects,” J. Opt. Soc. Am. 55, 1555-1556(1964), G. Cochran, “New method of making Fresnel transforms,” J. Opt.Soc. Am. 56, 1513-1517 (1966), P. J. Peters, “Incoherent holography withmercury light source,” Appl. Phys. Lett. 8, 209-210 (1966), H. R.Worthington, Jr., “Production of holograms with incoherentillumination,” J. Opt. Soc. Am. 56, 1397-1398 (1966), A. S. Marathay,“Noncoherent-object hologram: its reconstruction and opticalprocessing,” J. Opt. Soc. Am. A 4, 1861-1868 (1987), and G. Sirat, D.Psaltis, “Conoscopic holography,” Optics Letters, 10, 4-6 (1985), eachof which is incorporated herein by reference.

However, the conventional incoherent holographic techniques may requireimpractically high levels of light intensity. Thus, conventionalincoherent holographic systems require active illumination of objects,and therefore may exhibit the resulting problems and limitationsdescribed above.

In addition, conventional incoherent holographic systems may rely onilluminating the object with a bandwidth limited source to reducesensitivity to length differences in the plural optical pathdifferences. For example, in a conventional incoherent holographicsystem acceptable variations in the relative length of optical paths maybe limited to the inverse of the bandwidth multiplied by the lightvelocity. Thus, a predetermined light source having a limited bandwidthmay be required, and the elimination of extraneous illumination may benecessary in conventional incoherent holographic systems.

Further, conventional incoherent holographic systems may require opticalarrangements having plural optical axes similar to the example shown inFIG. 32. Thus, conventional incoherent holographic systems may also besusceptible to variations in direction or length of the optical axes,and attendant problems, as described above.

In addition, conventional holographic techniques involve splitting lightinto two channels using mirrors, which may have a low transferefficiency, and then recombining the split light. The efficiency may beparticularly low in the recombination where more than 50% of the powergets lost.

Further, in a conventional Fourier hologram, each object point istransformed to a linear grating throughout the entire image plane (e.g.,throughout an entire CCD plane). Thus, in conventional incoherentmethods of producing Fourier holograms, light from each object pointmust disadvantageously be intense enough to establish a high contrastgrating or pattern over the entire image plane.

Tomographic methods have been proposed to overcome the limitations ofconventional holographic techniques described above. Such tomographicmethods may involve capturing plural images of an object from differentpoints of view, for example by moving the object, or the camera, orboth, in between successive images, and extracting three-dimensionalobject information by processing the successive images. Conventionaltomographic methods are described in Y. Li, D. Abookasis and J. Rosen,“Computer-generated holograms of three-dimensional realistic objectsrecorded without wave interference,” Appl. Opt. 40, 2864-2870 (2001),and Y. Sando, M. Itoh, and T. Yatagai, “Holographic three-dimensionaldisplay synthesized from three-dimensional Fourier spectra of realexisting objects,” Opt. Lett 28, 2518-2520 (2003), each of which isincorporated herein by reference.

Tomographic methods may be slow, however, as they may require pluralimages to be captured before and after a relative perspective betweenthe object and camera is changed and thus may not be able to captureobjects which change during the time it takes to capture the multipleimages. Alternatively, tomographic methods may require more expensive orphysically large equipment that includes the ability to simultaneouslycapture images of the object from more than one perspective. Further,the methods may be impractical for distant objects or immovable objectsfor which it may be difficult to change a relative perspective from thecamera. In addition, if the object is moving in an unpredictable way, itmay be difficult to extract information from successive images withouthaving another source of information regarding the shape or themovements of the object.

Range-finding methods involve measuring the distance between anapparatus and various points on a surface of an object, and constructingan image or model of the object based on the distances. Further, somerange-finding methods may include a predetermined or controlled movementof the object or the apparatus to predictably change the view of theobject from the apparatus over time. Thus, the conventionalrange-finding methods may also include the predictable change in theview of the object to determine an exterior three-dimensional shape ofthe object. Conventional range-finding methods include systems thatilluminate points on an object with a laser, and measuring the amount oftime required for the laser light to travel between the object and thelaser source to determine a distance to the object based on the traveltime. Related methods include “painting” an object with a laser stripeor grid and examining a deformation in the observed grid to determinegeometric information of the object.

However, such range-finding methods require active illumination of theobject by a coherent laser, and therefore are not suitable forincoherently illuminated objects or for fluorescent or luminescentemitting objects. When coherent light sources can be used, they have theattendant problems and limitations of active illumination and coherentlight sources described above. Further, the methods may be difficult toperform if the object is moving in an unpredictable way, or if theobject is very close to the laser source.

Other range-finding methods may include using a camera with a lenshaving a narrow depth of field and a calibrated automatic focusingsystem. The automatic focus system automatically adjusts the lens tofocus on portions of the object, for example, by maximizing contrast inresulting images. Then, a range to the object is determined based on amechanical position of the calibrated lens. However, such a calibratedfocus technique may not be useful for objects having minimal opticalcontrast, or when objects or the apparatus is in motion. Further, theaccuracy of such a system may be limited by the mechanical tolerances ofthe calibrated lens.

Another conventional method includes extracting an object distance fromshadows in an image. For example, a conventional shadow method includescapturing an image of shadows produced when electromagnetic radiation(e.g., x-ray radiation or light) from an object is blocked by a maskwith a concentric ring pattern such as a Fresnel Zone Pattern (FZP,a.k.a., Transmission Zone Plate, Zone Plate, Zone Pattern, Fresnel ZonePlate, etc. . . . ) placed between the object and an image plane.

For the purposes of this application, an FZP may be understood to be atwo-dimensional pattern of alternating light and dark concentric ringsin which a thickness (e.g., radial width) of successive rings isinversely proportional to the distance from the center of the rings. Forexample, the nth ring of an FZP may transition (i.e., from dark to lightor light to dark) at a radius r described by the following equation (orby an approximation thereof):

r _(n)=√{square root over (nfλ)}

where n is an integer, λ is the wavelength of the applied light and f isthe focal length of the FZP.

When used with scattered point light sources, such as stars, therelative positions of the centers of the shadows in the image may beextracted from the image, and distances to the corresponding point lightsources may be calculated from the shadow center locations in the image.Such a method is described in L. Mertz and N. O. Young, “Fresneltransformations of images,” in Proceedings of Conference on OpticalInstruments and Techniques, K. J. Habell, ed. (Chapman and Hall, London1961) p.305, incorporated herein by reference.

However, such conventional shadow ranging methods have a limitedusefulness for when the captured electromagnetic radiation has awavelength comparable to the distance between the rings of the FZP, suchas visible light. For example, the visible light may be diffracted bythe edges of rings in the FZP causing shadows in the image to havepoorly defined or smeared edges, thereby making it difficult orimpossible to isolate the centers of resulting shadow patterns (e.g.,see Mertz and Young at FIG. 2).

Conventional scanning holographic methods involve scanning an object byilluminating a surface of the object with a moving a pattern of FresnelZone Plates (FZPs), serially sensing the intensity of reflected ortransmitted light from the object (i.e., a one dimensional intensitysignal) as the pattern moves across the object, and integrating andprocessing the serially sensed light intensities to generatethree-dimensional information of the object. In particular, aconvolution between the object and the moving Fresnel Zone Patterns isused to extract three-dimensional information regarding the object.Conventional scanning holographic methods are described in Poon T.-C.,“Three-dimensional image processing and optical scanning holography,”ADVANCES IN IMAGING AND ELECTRON PHYSICS 126, 329-350 (2003), and G.Indebetouw, A. El Maghnouji, R. Foster, “Scanning holographic microscopywith transverse resolution exceeding the Rayleigh limit and extendeddepth of focus,” J. Opt. Soc. Am. A 22, 892-898 (2005), each of which isincorporated herein by reference.

However, conventional scanning holographic methods require that apattern be moved across the object while the location of the object isfixed, thereby limiting the usefulness of the method. Alternatively, thepattern may be fixed and the object moved across the pattern, resultingin similar limitations.

In addition, the scanning process may be a relatively slow processrequiring mechanical movements. Thus, scanning is susceptible toproblems produced by mechanical deterioration and inaccuracy, such asreduced resolution as described above.

Further, since scanning holographic methods serially capture aone-dimensional light intensity signal from the object and integrate theserial signal to extract three-dimensional information, such systems arehighly susceptible to variations in the relative positions of thescanning apparatus and the object over the duration of the scan. Forexample, if such a system captures a first intensity during a first partof the scan, and a second intensity during a second part of the scan,any variation in the relative positions of the object and the scanningapparatus (or even minor changes in the internal arrangement of elementsof the scanning apparatus) may adversely introduce variations into thecaptured second intensity, thereby reducing an accuracy, resolution andusefulness of the system.

In addition, scanning holographic systems are generally large andcomplex therefore they may not be suitable for applications requiringportability or low cost. Further, conventional scanning holographicsystems may require an object to be illuminated by an interferencepattern produced by interfering laser light, may include a very slowrecording process that could take several minutes or more for eachobject capture, and the recording process may disadvantageously requiresignificant mechanical movement of recording device components and/orthe object during the recording process.

Holograms recorded by incoherent light open many new applications likeoutdoor and astronomical holography (J. B. Breckinridge,“Two-Dimensional White Light Coherence Interferometer,” Appl. Opt. 13,2760 (1974)) and fluorescence holographic microscopy (G. Indebetouw, A.El Maghnouji, R. Foster, “Scanning holographic microscopy withtransverse resolution exceeding the Rayleigh limit and extended depth offocus,” J. Opt. Soc. Am. A 22, 892-898 (2005)). The oldest methods ofrecording incoherent holograms have made use of the property that everyincoherent object is composed of many source points each of which isself spatial coherent and therefore can create an interference patternwith light coming from the point's mirrored image. Under this generalprinciple there are various types (J. B. Breckinridge, “Two-DimensionalWhite Light Coherence Interferometer,” Appl. Opt. 13, 2760 (1974)) (A.W. Lohmann, “Wavefront Reconstruction for Incoherent Objects,” J. Opt.Soc. Am. 55, 1555-1556 (1965)) (G. Sirat, D. Psaltis, “Conoscopicholography,” Optics Letters, 10, 4-6 (1985)) of holograms includingFourier (J. B. Breckinridge, “Two-Dimensional White Light CoherenceInterferometer,” Appl. Opt. 13, 2760 (1974)) (G. W. Stroke and R. C.Restrick, “Holography with Spatially Incoherent Light,” Appl. Phys.Lett. 7, 229 (1965)) and Fresnel holograms (G. Cochran, “New method ofmaking Fresnel transforms,” J. Opt. Soc. Am. 56, 1513-1517 (1966)) (P.J. Peters, “Incoherent holography with mercury light source,” Appl.Phys. Lett. 8, 209-210 (1966)). The process of beam interfering demandshigh levels of light intensity, extreme stability of the optical setupand relatively narrow bandwidth light source. These limitations haveprevented holograms from becoming widely used for many practicalapplications.

More recently two groups of researchers have proposed to computeholograms of 3-D incoherently illuminated objects from a set of imagestaken from different points of view. (Y. Li, D. Abookasis and J. Rosen,“Computer-generated holograms of three-dimensional realistic objectsrecorded without wave interference,” Appl. Opt. 40, 2864-2870 (2001))(Y. Sando, M. Itoh, and T. Yatagai, “Holographic three-dimensionaldisplay synthesized from three-dimensional Fourier spectra of realexisting objects,” Opt. Lett 28, 2518-2520 (2003)) This method, althoughit shows promising prospects, is relatively slow since it is based oncapturing tens of images of the scene images from different view angles.

Another method is called scanning holography (G. Indebetouw, A. ElMaghnouji, R. Foster, “Scanning holographic microscopy with transverseresolution exceeding the Rayleigh limit and extended depth of focus,” J.Opt. Soc. Am. A 22, 892-898 (2005)) (Poon T.-C., “Three-dimensionalimage processing and optical scanning holography,” Adv. in Imag. & Elec.Phys. 126, 329-350 (2003)) in which a pattern of Fresnel Zone Plate(FZP) scans the object such that at each and every scanning position thelight intensity is integrated by a point detector. The overall processyields a Fresnel hologram obtained as a convolution between the objectand FZP patterns. However the scanning process is relatively slow and isdone by mechanical movements. A similar convolution is actually donealso in the present work; however, unlike the case of scanningholography, we propose here a convolution without movement.

Mertz and Young (L. Mertz and N. O. Young, “Fresnel transformations ofimages,” in Proceedings of Conference on Optical Instruments andTechniques, K. J. Habell, ed. (Chapman and Hall, London 1961) p.305)already proposed holographic photography based on convolution withoutmovement between object and FZPs. However, their process relies ongeometrical optics, which cannot yield good imaging results in theoptical regime. On the contrary, our suggested correlator forimplementing the holographic recording is valid in the optical regime,since its operation principle is based on the diffraction theory (J.Goodman, Introduction to Fourier Optics, 2^(nd) ed., McGraw-Hill, NewYork, 1996, pp. 63-95 (Chapter 4).

Referring now to the drawings, wherein like reference numerals designateidentical or corresponding parts throughout the several views.

FIG. 1 is a block diagram of an optical apparatus 100 according to afirst embodiment of the present invention. The optical apparatus 100 isconfigured to capture a three-dimensional information of an object 130,and the optical apparatus 100 includes an optical assembly 110 and animage capture assembly 120. In particular, the optical assembly 110receives light from the object 130 along a receiving optical axis 140.For example, the optical assembly may receive light from the sun 150that is reflected or scattered by reflecting surfaces on the object 130.The received light may be polychromatic and incoherent light, such asreflected sunlight, or may also include monochromatic light or coherentlight. In addition, the light from the object may be fluorescent lightor chemiluminescent light emitted by the object. The optical apparatus100, in this embodiment, does not illuminate the object but passivelyreceives light from the object.

The optical assembly 110 transforms the received light according to atransformation described below, and transmits the transformed lightalong the receiving optical axis 140. The image capture assembly 120receives the transformed light from the optical assembly 110, andcaptures a two-dimensional intensity image of the transformed light. Thecaptured two-dimensional intensity image includes three-dimensional orgeometric information regarding the portions of the object 130 fromwhich light is received at the optical assembly 110. Thethree-dimensional or geometric information is encoded in the capturedtwo-dimensional intensity image as a Fresnel hologram. In other words, ahologram, as discussed in this specification, is a two-dimensional imagethat encodes three-dimensional information. In addition, the presentinvention also applies to capturing a volume hologram, which is athree-dimensional intensity image that encodes three-dimensional orgeometric information of an object. The image capture assembly 120 mayextract the three-dimensional information from the captured image. Theimage capture assembly may be an opaque light capturing device. Anopaque device is understood to mean a device that is not transparent ortranslucent to electromagnetic radiation of relevant frequencies andintensities, and therefore such a device does not allow suchelectromagnetic radiation to pass through.

A Fresnel hologram is a real positive light intensity distribution thatencodes a complex valued wave-front distribution, includingthree-dimensional information regarding the light scattering surface ofthe object. Further, in a Fresnel hologram, each point on the object isencoded into a portion of a sinusoidal Fresnel zone plate with an entirerange of spatial frequency components present, as noted by Goodman,“Introduction to Fourier Optics,” 3rd Ed., Roberts & Company Publishers,2005, incorporated herein by reference. Thus, a three-dimensional imageof the object may be recreated optically, by appropriately illuminatinga transparency having the Fresnel hologram, or the three-dimensionalimage of the object may be recreated by a computer using an electronicimage data of the Fresnel hologram. The recreated three-dimensionalimage of the object includes three-dimensional information regarding theshape and distance of an observable surface of the object.

The optical apparatus 100 may advantageously capture thethree-dimensional object information without moving or being moved(i.e., the spatial relationship between the optical apparatus 100 andthe object 130 may remain unchanged from a time before an image iscaptured to a time after the three-dimensional information is extractedfrom the captured image by the image capture assembly 120). In addition,the optical apparatus 100 may advantageously capture thethree-dimensional object information while one or both of the object andthe optical apparatus 100 are in motion.

Moreover, the optical apparatus 100 does not project any pattern on theobject, such as is done in a conventional or scanning holographicmethod. Further, the optical apparatus 100 does not include any partsthat are required to move while the light is being received from theobject, such as a scanning aperture used in scanning holography. Thus,without parts that move during image capture, the optical apparatus 100may be less expensive to produce and use, more reliable, quieter, andfaster, for example, than an apparatus used for scanning holography.Further, with respect to conventional holographic systems that requireactive illumination (for example, illumination by a laser), the presentinvention advantageously has a simpler design that may be applied tomore types of imaging.

In addition, the present invention does not require an interferencebetween a light from the illumination (i.e., not scattered by theobject) with a light scattered by the object. Instead, the currentapproach diffracts light scattered by the object, which may beunderstood as a mutual interference between portions of electromagneticradiation wavefronts coming from object itself, and is not aninterference between such scattered light and another light from thesource. Thus, as the mutual interference may be performed by a fewcollinear electromagnetic elements (e.g., lenses and masks, or DOEs, asdescribed below), or even a single electromagnetic element (e.g., asingle DOE, as described below), the relative differences betweenoptical paths of the interfering wavefronts are easily controlled (e.g.,all the optical paths pass through the same electromagnetic elements)and therefore, variations between the lengths of the paths may be moreeasily controlled and minimized.

Further, the optical apparatus 100 may advantageously capture thethree-dimensional object information in a single image (e.g., a singleexposure).

Moreover, the optical apparatus 100 may advantageously be able tocapture images with very low levels of light intensity. Conventionalholographic systems may require beam splitters and/or mirrors that maycause some received light to be lost or wasted. On the other hand, theoptical apparatus 100 does not require the use of beam splitters ormirrors, and therefore may be able to capture images with low levels oflight intensity.

Further, conventional holographic systems may produce Fourier hologramsin which each object point contributes to interference fringe patternsthat are spread over the entire image plane. Such conventional systemsmay require greater light intensity than the optical apparatus 100,which translates each object point using a Fresnel Zone Pattern, whichmay produce fringe patterns for a particular object point in only aportion of the image plane, thereby advantageously allowing for lowerlight intensities.

In addition, the optical apparatus 100 advantageously receives andtransmits light only along a single axis, thereby reducingsusceptibility to axial variation and simplifying the design,manufacture and use of the optical apparatus 100. Further, the opticalapparatus 100 is coaxial and self-interfering. In particular, in thepresent embodiment light from separate light paths is not interfered toproduce an interference pattern or hologram. Instead, the hologram isproduced by diffraction of light in the optical assembly 110. Althoughdiffraction may be understood as being produced by interference betweeneach portion of a light wavefront (i.e., each point along the wavefrontbeing considered a point source of light according to Huygens wavetheory of light), diffraction produced by a single coaxial assembly, asin the present embodiment, is much less sensitive to variations inoptical paths between interfering light sources. In particular, light isself-interfered, according to the present embodiment, because the onlyinterference is between light waves passing through various portions ofa same optical element (e.g., the optical assembly 110, or the mask 304in FIG. 3, described below), and it is much easier to minimize pathlength and angle variations for paths passing through a same opticalelement, as in the present embodiment, than it is to control pathvariations between separate optical paths, passing through separateoptical elements, along separate optical axes, as in the conventionalholography systems. In addition, the optical apparatus 100 may be usedto advantageously capture polychromatic incoherent light received fromthe object. Therefore, a full color three-dimensional image may berecreated from the Fresnel hologram recorded by the apparatus.

Although embodiments within this document are described as transmittingand receiving light, capturing light images and including opticalassemblies, the invention is also applicable to other types ofelectromagnetic radiation. Thus, the invention also includes anelectromagnetic radiation apparatus that includes an electromagneticradiation assembly that receives a received electromagnetic radiationfrom an object.

FIG. 2 is a block diagram that illustrates an example of capturedgeometric information according to embodiments of the present invention.According to the example of FIG. 2, an object 200 is illuminated by alight source or sources (e.g., the sun 150) causing light to bescattered or reflected by various light radiating portions of the object200. Three example light radiating portions 206, 208 and 210 scatterlight rays 216, 218 and 220, respectively. These example light raystravel towards the optical apparatus 100 (shown in FIG. 2 without thedetail of FIG. 1). Light captured by the optical apparatus 100,according to the present invention, includes geometric informationregarding the distance between the object and the optical apparatus aswell as the shape of observable surfaces of the object 200 from whichlight is received at the optical apparatus 100. For example, thecaptured light includes information regarding a distance traveled by theray of light 218, and in particular, includes the distance between thelight radiating portion 208 and the optical apparatus 100. Further, thecaptured light also includes geometric information regarding ahorizontal distance of the light radiating portion 208, for example, ahorizontal distance 212 between an edge of the object 200 and the lightradiating portion 208. In addition, the captured light also includesgeometric information regarding a vertical distance of the lightradiating portion 208, for example, a vertical distance 214 between anedge of the object 200 and the light radiating portion 208. In thisexample, horizontal distance 212 and vertical distance 214 are distancesmeasured in a measurement plane 204 that passes through radiatingportion 208.

Thus, an optical apparatus, according to the present embodiment, may beconfigured to capture a light including geometric information regardingeach portion of each object from which the light is received at theoptical apparatus. Further, from the geometric information, the size,shape and location of the visible portions of each object may bedetermined. For example, in FIG. 2, if light is scattered by eachexternal surface of the object 200, and at least a portion of thescattered light is received at the optical apparatus 100, then theapparatus 100 may capture light including geometric informationregarding the dimensions (e.g., height, width and depth) of each visiblesurface of the object 200, as well as information regarding the distancebetween the object 130 and the front surface of optical apparatus 100.

Although light is scattered by external surfaces in FIG. 2, one of skillin the art will understand that such an optical apparatus is alsocapable of capturing received light from an internal surface of theobject 130 that radiates light to the optical assembly 100 through atranslucent or transparent exterior portion of the object 130. In thatcase, the captured geometric information may include geometricinformation regarding an interior portion of the object.

The optical assembly 110 includes any optical assembly configured tocontrol a complex amplitude of the transmitted light according to thecomplex transformation function described below. Thus, for example,optical assembly 110 may include one or more refractive lenses, one ormore diffractive optical elements (DOEs), or one or more spatial lightmodulators (SLMs).

An incoherent correlator in the regime of diffraction theory may includeevery system that produces a pattern of a Fourier transform of the masktransparency on the system's aperture at an output plane around a pointthat is linearly related to an input point's location, when theincoherent correlator is illuminated by a single point from someposition on the input plane. Thus, the incoherent correlator produces anoutput image including every point in the input plane.

FIG. 3 is a block diagram of an incoherent correlator 300 that may beused as the optical assembly 110 in the optical apparatus 100 shown inFIG. 1. The incoherent correlator 300 is an optical assembly thatincludes a first transforming optical assembly 302, a mask 304 and asecond transforming optical assembly 306. Each of the first and secondtransforming optical assemblies 302/306 include the types of converginglenses that would perform a two-dimensional Fourier transform ofreceived light if they were illuminated by coherent light (thoughcoherent light is not required during the operation of the presentinvention). When the incoherent correlator 300 is illuminated by asingle point light source 308 from some position on a plane 318, theincoherent correlator 300 produces a pattern of a Fourier transform of amask 304 on an output plane 316. The plane 318 is located along andperpendicular to an optical axis 320 of the incoherent correlator 300,at a distance (z+f₁) from the first transforming optical assembly 302,where f₁ is a focal length of the first transforming optical assembly302 and z is a remaining distance between the point light source 308 andthe first transforming optical assembly 302. The output plane is locatedalong and perpendicular to the optical axis 320 at a distance f₂ fromthe second transforming optical assembly 306, where f₂ is a focal lengthof the second transforming optical assembly 306, in a direction awayfrom the plane 318. Note that the Fourier transform of a mask 304 on anoutput plane 316 is obtained only if z=0. When z≠0 the optical assembly300 still performs a correlation between the object, but with adifferent function than the Fourier transform of mask 304. In otherwords, in the case that z≠0, the output plane 316 will include an outputimage that is different than a Fourier transform of a mask 304.

FIG. 4 is a block diagram of an embodiment of an optical apparatus 400that includes an optical assembly 300. Optical apparatus 300 includes afirst transforming optical assembly 302, a mask 304 and a secondtransforming optical assembly 306. Each of the first and secondtransforming optical assemblies 302/306 are the types of opticalassemblies (e.g., converging Fourier lenses) that would perform atwo-dimensional Fourier transform operation on a received coherent light(although the use of the apparatus does not require coherent light).Light is received from object 130 at the first transforming opticalassembly 302, which transforms the received light and transmits thetransformed light. The mask 304 receives the transformed light, variesan amplitude and/or phase of the received coherent transformed light asdescribed below, and transmits a portion of the received light as themasked light. The masked light is received by the second transformingoptical assembly 306, which transforms the masked light and transmits asecond transformed light. The image capture assembly 120 receives andcaptures the second transformed light, as described above. Note thatwhen the light received from the object is incoherent, thetransformations performed by the first and second transforming opticalassemblies 302/306 may not be Fourier transformations. However, thefirst and second transforming optical assemblies 302/306 are the typesof optical assemblies (e.g., converging Fourier lenses) that wouldproduce a Fourier transform of received coherent light or received pointsource light.

The mask 304 includes any device or structure configured to transform anamplitude and phase of a received light, according to one or morepredetermined complex transmission functions. For example, the mask 304may include one or more diffractive optical elements (DOE), one or moreamplitude filters, one or more lenses, and/or one or more SLMs.

FIG. 5A is a detailed front view of an embodiment of mask 304 thatincludes a DOE having an array of plural transform regions 500-514. Eachof the plural transform regions in the diffractive optical element isconfigured to transform a phase and/or an amplitude of a received lightaccording to the transform equations described below.

The diffractive optical element may include volume-modulated diffractiveoptical elements that use a variation in the volume of refractivematerial in each transform region to transform the phase and/oramplitude of the received light and produce transformed transmittedlight. In addition, the diffractive optical element may includeindex-modulated diffractive optical elements that use a variation in arefractive index of refractive material in each transform region totransform the phase and/or amplitude of the received light and producetransformed transmitted light. In addition, the diffractive opticalelement may include one or more transmission layers of having apredetermined transmissivity to thereby vary an amplitude of thereceived light. Further, diffractive optical elements that combine oneor more features of volume-modulated, index-modulated and transmissionlayer diffractive optical elements may also be included. Methods ofpreparing the diffractive optical elements include, for example,conventional methods such as those described in Salmio et al.,“Graded-index diffractive structures fabricated by thermal ionexchange,” Applied Optics, Vol. 36, No. 10, 1 Apr. 1997, Carre et al.,“Customization of a self-processing polymer for obtaining specificdiffractive optical elements,” Synthetic Metals 127 (2002) 291-294, andNordman et al., “Diffractive phase elements by electron-beam exposure ofthin As₂S₃ films, Journal of Applied Physics 80(7), 1 Oct. 1996, each ofwhich is incorporated herein by reference.

FIGS. 5B-5G show a view of cross-section AA′ in FIG. 5A for variousembodiments of optical assembly 110. FIG. 5B is a cross-section view ofa symmetrically arranged volume modulated diffractive optical elementstructure, in which the volume of transform regions 500-514 aresymmetrical with respect to a center line 516.

In FIGS. 5C-5G, each of the transform regions corresponds to a transformregion in the embodiments of FIGS. 5A and 5B (e.g., transform regions500C, 500D, 500E, 500F and 500G correspond to transform region 500),however, with the properties of the relevant embodiment.

FIG. 5C is a cross-section view of a volume modulated diffractiveoptical element structure in which the volume of transform regions500C-514C are varied and arranged asymmetrically with respect to acenter line 516.

FIG. 5D is a cross-section view of an index modulated diffractiveoptical element structure in which a refractive index of transformregions 500D-514D is varied.

The optical assembly 110 is not limited to embodiments including onlyvolume or index modulated transform regions, but also includes a mixtureof volume and index modulated transform regions, as well as transformregions that include both volume and index modulation features.

FIG. 5E is a cross-section view of a mixed mode diffractive opticalelement structure in which transform regions 500E, 506E, 508E and 512Einclude varied refractive indexes, transform regions 502E, 510E and 514Einclude varied volumes and transform region 504E includes both a variedrefractive index and a varied volume, for example.

The optical assembly 110 is not limited embodiments including only arefractive-type diffractive optical assembly, in which light passes fromone side of the assembly to exit at another side, but also includesembodiments having a reflective-type diffractive optical assembly, inwhich light is reflected by a surface prior to exiting the assembly.

FIG. 5F is a cross-section view of a reflective volume modulateddiffractive optical element including a reflective layer 518 and volumemodulated transform regions 500F-514F. In this embodiment, the receivedlight enters through transform regions 500F-514F, is reflected byreflective layer 518 and passes again through transform regions500F-514F before exiting the diffractive optical element.

The optical assembly 110 is not limited to embodiments includingtransform regions having the shapes shown in the figures above, but alsoincludes embodiments in which the transform regions have other shapes,for example a rounded shape.

FIG. 5G is a cross-section view of a reflective volume modulateddiffractive optical element in which the transform regions 500G-514Ghave a different shape on an external edge and a square shape on aninternal edge adjacent to the reflective layer 518.

FIG. 6A is a detailed front view of an embodiment of mask 304 thatincludes a DOE having an array of plural transform regions 600-614, eachof which may include features similar to those described above regardingFIGS. 5A-5G. Further, each of the plural transform regions in thediffractive optical element of this embodiment is configured totransform an amplitude of a received light using one or moretransmission layers configured to reduce an amplitude of received lightby various predetermined amounts according to the transmission equationsbelow. In this embodiment, transform region 614 is configured to includea transmission layer including a material such as an absorbent ink or areflective metal configured to reduce an amplitude or intensity oftransmitted light by a relatively small amount, while transform regions606, 610 and 602 are configured to reduce an amplitude or intensity oftransmitted by respectively increasing amounts. The variedtransmissivity may be produced by varying a number of layers of a samematerial, varying a density of a same material, mixing variousconcentrations of materials, or by any conventional method used to varyan intensity of a received light.

Further, each transform region 600-614 may be configured to applydifferent amounts of amplitude or intensity reduction over differentfrequencies of the received light spectrum. Thus, each transform region600-614 may include an ability to differently filter each color of thereceived light. For example, transform region 606 may transmit a portionof received light having frequencies close to the color red withoutreduction in amplitude and may reduce the amplitude of all otherfrequencies of the received light. Further, transform region 610 mayreduce the amplitude of received blue light by a first amount and mayreduce the amplitude of a received yellow or red light by a secondamount. All permutations of frequency attenuating profiles for eachtransform region are included in the invention.

FIGS. 6B-6E show views of cross-section BB′ in FIG. 6A for variousembodiments of mask 304. FIG. 6B is a cross-section view of a maskincluding a transmission layer 620 having varied transmissivity forregions adjacent to corresponding transform regions 514F, 506F, 510F and502F.

FIG. 6C is a cross-section view of a mask having transform regions 614C,606C, 610C and 602C configured to vary an amplitude of the receivedlight (e.g., for example, due to impurities added to the transformregion)

FIG. 6D is a cross-section view of a mask having a printed overlay 622attached to one side. The printed overlay 622 includes printed regions624 in which ink or other light absorbing or reflecting material isdeposited in regions adjacent to corresponding transform regions 614D,606D, 610D and 602D in varying concentrations or amounts to vary anamplitude of the received light. The printed overlay 622 may be printedprior to being attached to the rest of the mask 304. Further, theprinted regions 624 may be printed directly on the DOE without using aprinted overlay 622, as shown in the mask 304 embodiment in FIG. 6E.

The DOE in the mask is not limited to DOEs having an array of 64transform regions as shown in the examples above, but also includes DOEshaving any other number of transform regions, and includes transformregions arranged other than in an array, such as with a radialarrangement of transform regions (not shown). Further the transformregions may have any shape and be in any arrangement.

The mask 304 may be configured to simultaneously perform one or morecomplex transmission functions. In the present embodiment, the maskincludes three different transmission functions to produce threedifferent convoluted and transformed partial images within each capturedimage. When particular transmission functions, as described below, areincluded in the mask 304, the resulting three different convoluted andtransformed partial images in the captured image may be combined as aFresnel hologram of the object 130, from which three-dimensional objectinformation may be extracted.

FIG. 6F is a block diagram of an embodiment of mask 304 including anamplitude SLM 628 that is configured to controllably modify an amplitudeof a received light, and phase SLM 630 that is configured tocontrollably modify a phase of the amplitude modified light and that ismounted adjacent to the amplitude SLM 628.

FIG. 6G is a block diagram of an alternative embodiment of mask 304 inwhich amplitude SLM 628 and phase SLM 630 are not located next to eachother, but have an intervening space. Further, the SLMs mayalternatively be placed in a different order with respect to the lightpath (not show). Further, the invention may include other SLMs that maybe available now or in the future and are configured to controllablymodify both the amplitude and phase of a received light.

The two-dimensional intensity image captured by the image captureassembly is generally described by an intensity function o(x,y), whichdescribes the distribution of light intensities captured at each pointin the image capture plane (i.e., x, y plane). The three intensityfunctions o_(n)(x,y) define the partial contribution to the overallimage intensity contributed by each partial image, and is related to theoverall image intensity function as follows:

$\begin{matrix}{{o\left( {x.y} \right)} = {\sum\limits_{n = 1}^{3}{B_{n}{o_{n}\left( {x,y} \right)}}}} & \left( {1A} \right)\end{matrix}$

where B_(n) is a complex constant.

The transmission functions that produce the three partial imagescaptured by the image capture assembly 120 are defined as follows:

o _(n)(x,y)=∫∫∫s(x′,y′,z′)|h _(n)(x−x′,y−y′,z′)|² dx′dy′dz′  (1B)

where s(x′,y′,z′) is a function that describes the intensity at thesystem input in the vicinity of the point (x′,y′,z′)=(0,0,0). From thefunction o(x,y), the geometric information regarding the lightscattering surface (i.e., the portions of the object facing the opticalapparatus 100 that scatter or emit light that is received at the opticalassembly 110) of the object may be determined in terms of objectreferenced coordinates x′, y′ and z′.

The transformed light includes point spreading functions (PSF)h_(n)(x,y,z) contributed by each transmission function. In the presentembodiment, h(x,y,z) is a linear summation of point spreading functionsh₁(x,y,z), h₂(x,y,z) and h₃, which each perform a light spreadingfunction with respect to the image capture coordinates (i.e., x, y, z).PSF h(x,y,z) is defined as follows:

$\begin{matrix}{{{h\left( {x,y,z} \right)} = {\sum\limits_{n = 1}^{3}{\left( {{\frac{1}{\sqrt{2}}\exp \left\{ {{\frac{\pi}{2{{\lambda\Delta}(z)}}\left\lbrack {\left( {x - x_{n}} \right)^{2} + \left( {y - y_{n}} \right)^{2}} \right\rbrack} + \frac{{\theta}_{n}}{2}} \right\}} + {\frac{1}{\sqrt{2}}\exp \left\{ {{\frac{- {\pi}}{2{{\lambda\Delta}(z)}}\left\lbrack {\left( {x - x_{n}} \right)^{2} + \left( {y - y_{n}} \right)^{2}} \right\rbrack} - \frac{{\theta}_{n}}{2}} \right\}}} \right){p_{z}\left( {{x - x_{n}},{y - y_{n}}} \right)}}}},} & (2)\end{matrix}$

where p_(z)(x−x_(n),y−y_(n)) are two-dimensional disk functions centeredat points (x₁,y₁), (x₂,y₂) and (x₃,y₃), respectively, in the imagecapture space, i is the imaginary unit (i.e., i=(−1)^(0.5)), λ is thewavelength of the propagating light, and Δ(z) is a parametermonotonically related to the distance z. Further, the disk functionp_(z) has a diameter function d(z) that varies the diameter based on thevalue of z and thereby limits the diameter of a corresponding FZP.Further, each PSF is selected to have a different constant phase valueθ_(n).

Although the equation above includes a single value λ for the wavelengthof the propagating light, the above equation may be used forpolychromatic light by assuming that the captured intensity image is acombination of intensities in plural portions of the total capturedspectrum, and for example, the captured intensity image may beconsidered as a combination of captured red light intensities, capturedyellow light intensities, and captured green light intensities. Further,the invention also includes using other color models to represent thecolored image, such as CMYK.

Thus, in the image captured at the image capture device (i.e., at z=0),the PSFs h_(1,2,3) are given by

$\begin{matrix}{{{h\left( {x,y,0} \right)} = {\sum\limits_{n = 1}^{3}{\left( {{\frac{1}{\sqrt{2}}\exp \left\{ {{\frac{\pi}{2{{\lambda\Delta}(0)}}\left\lbrack {\left( {x - x_{n}} \right)^{2} + \left( {y - y_{n}} \right)^{2}} \right\rbrack} + \frac{{\theta}_{n}}{2}} \right\}} + {\frac{1}{\sqrt{2}}\exp \left\{ {{\frac{- {\pi}}{2{{\lambda\Delta}(0)}}\left\lbrack {\left( {x - x_{n}} \right)^{2} + \left( {y - y_{n}} \right)^{2}} \right\rbrack} - \frac{{\theta}_{n}}{2}} \right\}}} \right){p_{o}\left( {{x - x_{n}},{y - y_{n}}} \right)}}}},} & (3)\end{matrix}$

Therefore, the desired light transforming function of each partialfunction H_(n)(u,v) of the optical assembly 110 is the Fourier transformof h_(n)(x,y,0) in equation 3 above. Thus, H_(n)(u,v), corresponding toembodiments with the spatial multiplexing of partial mask patterns asshown in Equation 3 and FIGS. 9A-9C, 10A and 10B, is defined as follows:

$\begin{matrix}{{{H_{n}\left( {u,v} \right)} = {\left\{ {{\frac{1}{2}{\exp \left\lbrack {{\frac{\pi}{{\lambda\gamma}_{1}}\left( {u^{2} + v^{2}} \right)} + {\frac{2\pi}{\lambda \; f_{2}}\left( {{x_{n}u} + {y_{n}v}} \right)} + \frac{{\theta}_{n}}{2}} \right\rbrack}} + {\frac{1}{2}{\exp \left\lbrack {{\frac{\pi}{{\lambda\gamma}_{2}}\left( {u^{2} + v^{2}} \right)} - {\frac{2\pi}{\lambda \; f_{2}}\left( {{x_{n}u} + {y_{n}v}} \right)} - \frac{{\theta}_{n}}{2}} \right\rbrack}}} \right\}*{P\left( {u,v} \right)}}},{n = 1},2,3} & \left( {4A} \right)\end{matrix}$

In alternative time multiplexing embodiments, such as those shown inFIGS. 15, 17 and 18, H_(n)(u,v) may be defined as follows:

$\begin{matrix}{{{H_{n}\left( {u,v} \right)} = {\left\{ {{\frac{1}{2}{\exp \left\lbrack {{\frac{\pi}{{\lambda\gamma}_{1}}\left( {u^{2} + v^{2}} \right)} + \frac{{\theta}_{n}}{2}} \right\rbrack}} + {\frac{1}{2}{\exp \left\lbrack {{\frac{\pi}{{\lambda\gamma}_{2}}\left( {u^{2} + v^{2}} \right)} - \frac{{\theta}_{n}}{2}} \right\rbrack}}} \right\}*{P\left( {u,v} \right)}}},} & \left( {4B} \right) \\{\mspace{79mu} {{n = 1},2,3}} & \;\end{matrix}$

where * represents a convolution operation.

Further, the overall transforming function of the mask H(u,v) is definedas follows:

$\begin{matrix}{{H\left( {u,v} \right)} = {\sum\limits_{n = 1}^{3}{H_{n}\left( {u,v} \right)}}} & (5)\end{matrix}$

where u and v are coordinates in the plane of the optical assembly 110corresponding to the x and y coordinates in the image capture plane(i.e., the u axis is parallel to the x axis and the v axis is parallelto the y axis), and P(u,v) is the Fourier transform of disk functionp_(o)(x,y).

In the equations above, y_(1,2) may be defined according to thefollowing equation:

$\begin{matrix}{\gamma_{1,2} = {\pm \frac{27f_{1}^{2}}{4f_{2}}}} & (6)\end{matrix}$

Note, however, that the invention is not limited to using opticalassemblies having a different focal length and that the equations may beextended to allow same focal lengths for the two optical assemblies. Inother words, the parameters of mask y_(1,2)=±y are determined accordingto Equation 6 when the parameters f₁ and f₂ are known. f₁ and f₂ may bechosen according to the resolution of the SLM or DOE For example, if theSLM area is D×D, with N×N pixels, then the pixel size is δ=D/N. Theminimal ring width of FZP displayed on this SLM is well known asδ=|y|λ/D. Therefore, from the equation D/N=|y|λ/D one gets |y|=D²/Nλ.Then, substituting |y| into Equation 6 yields the values of theparameters f₁ and f₂ Note that this example illustrates only onepossible set of considerations, but the invention also includes methodsand resulting apparatuses produced with filter parameters calculatedbased on other factors.

When the transform regions of a mask include a color filteringcapability configured to filter out or to pass light only centered atparticular frequencies, as described above with respect to FIG. 6A. Themasks do not have to change when illuminated by various colors. Theresponse of a mask transparency changes according to the wavelength ofthe light, for example as shown in Equation 27 below.

FIG. 7 shows an example of a binary Fresnel Zone Pattern 700, in whicheach of zone includes only one of two transmissivity states:substantially transparent, and substantially opaque, with respect to thelight being transmitted. In this example, the FZP 700 is printed on aglass substrate 702 that transmits more than 90% of light within thevisual light spectrum using an ink that reflects or absorbs more than90% of light within the relevant light spectrum.

The invention is not limited to binary FZPs having alternating zones ofmore than 90% transmission and more than 90% absorption/reflection, butalso includes FZPs having other levels of transmission, andabsorption/reflection, as known in the field of FZPs. Further, theinvention is not limited to FZPs having zones having a consistenttransmissivity throughout each zone (i.e., zones that are entirelysubstantially transparent or entirely substantially opaque), but alsoincludes FZPs having zones with varying transmission levels within eachzone. In addition, the invention is not limited only to patterns ofcomplete circular rings, but also includes patterns of partial rings,such as an off-axis FZP. Moreover, the invention also includes replacingthe FZP with a photon sieve, such as described in Kipp et. al., “Sharperimages by focusing soft x-rays with photon sieves,” Nature, vol. 414, 8Nov. 2001, pp. 184-188, incorporated herein by reference.

FIGS. 8A-8C show examples of sinusoidal FZPs 800, 802 and 804. Insinusoidal FZPs, transmissivity varies sinusoidally between points ofmaximum transmissivity in substantially transmissive zones and points ofminimum transmissivity in less transmissive zones, along a straight lineradiating from the center of the FZP. Further, the FZPs 800, 802 and 804each have a different phase.

FIGS. 9A-9C show examples of Fourier Transformed FZP patterns (FT-FZP)900, 902 and 904 that are Fourier transforms of FZPs 800, 802 and 804,respectively. Note that the FT-FZP patterns may also have an FZPconcentric pattern. However, where the FZPs in FIGS. 8A-C produce animage having an intensity vs. radial coordinate distribution having auniform amplitude across the radial coordinates, the FT-FZPs in FIGS.9A-C have an intensity vs. radial coordinate distribution similar to abell curve with an intensity peak at the centers of the rings in thering patterns, and reduced intensities at all other locations. TheFourier transforms of Fresnel zone patterns may be used to produce themask functions according to Equation 4 above.

Further, H(u,v) of Equation 4 may be obtained by Fourier transform ofh(x,y,0) from Equation 3 multiplied by the quadratic phase function.Note that an FZP may be a sum of two quadratic phase functions withopposite signs in their arguments, and the Fourier transform of aquadratic phase function is also a quadratic phase function. Therefore,each quadratic phase of h(x,y,0) is multiplied by a quadratic phasefunction and then Fourier transformed to another quadratic phasefunction. The net result is that H(u,v) is a sum of two quadratic phasefunctions. It is possible to carefully choose h(x,y,0) to make sure thatH(u,v) will be a sum of two quadratic phase functions with argumentsthat are equal in their absolute value and have opposite signs. In thatcase, the sum of quadratic phase functions is a FZP with a bell curvebecause h(x,y,0) has a disk shape. In particular, a disk function may betransformed to what is called a Mexican-Hat function, which is convolvedwith the infinite FZP as shown in Equation 4. This convolution maygradually decrease the amplitude of the FZP as radius values increase,thereby creating the bell curve shape. According to the features ofFourier transforms the restricted area on h(x,y,0) causes a convolutionof H(u,v) with a narrow function indicated in Equation 4 by P(u,v). Thisconvolution is responsible for the bell-like shape of envelope ofH(u,v).

FIG. 10A is the amplitude portion of a complex transmission functionaccording to Equation 5, which is a linear combination of three maskfunctions each according Equation 4B, and corresponding to the Fouriertransform of the three FZPs in FIG. 10C.

FIG. 10B is the phase portion of the complex transmission functionaccording to Equation 5, which is a linear combination of mask functionsaccording to Equation 4, and corresponding to the Fourier transform ofthe three FZPs in FIG. 10C.

The complex transmission function of Equation 5 and examples illustratedin FIGS. 10A and 10B may be implemented using a single DOE or SLM orcombinations of DOEs and/or SLMs, as described above.

Note that the FT-FZPs (e.g., as shown in FIGS. 9A-C) may be an amplitudeonly real function, but the linear combination of FT-FZPs is anamplitude and phase pattern (e.g., as shown in FIGS. 10A-B). This ispossible by a careful choice of h(x,y,0). When h(x,y,0) is chosen to betwo particular quadratic phase functions that, when multiplied by thequadratic phase function of the lens and performing a Fourier transformof the resulting product, the obtained result is two quadratic phasefunctions having arguments that are equal in their absolute values, butwith opposite signs. In that case, the sum is a purely real function. Onthe other hand this property may not occur with the H(u,v) shown inFIGS. 10A-B, because the combination of 3 FZPs together is not symmetricin the sense that h(x,y,0)≠h(−x,−y,0). Further, it is well known that aFourier transform of non-symmetric functions can not be purely real.

FIG. 10C is an example of the pattern that is generated on the CCD whena point object is present at the input, as described above during theprocess to produce mask patterns shown in FIGS. 10A and 10B.

FIG. 11A is a block diagram of an image capture assembly 1100 thatincludes a light intensity capture device 1102 and a capture controlunit 1104. The light intensity capture device 1102 of this embodiment isa conventional light capturing device, such as a charge coupled device(CCD) as used in digital cameras, and is configured to capture atwo-dimensional array of light intensity information (i.e., image of thereceived light) under the control of the capture control unit 1104. Theinvention is not limited only to CCDs but may also include other devicesthat capture light intensity, such as a photographic film or atransparent film, an X-ray detector, other electromagnetic radiationdetectors, a CMOS device, a diode array, or a photo-detector, etc. . . ..

The capture control unit 1104 controls the light capturing functions ofthe light intensity capture device 1102 and is configured to retrieveelectronic image data information from the light intensity capturedevice 1102. For example, in the present embodiment, the light intensitycapture device includes a CCD connected to the capture control unit1104, which is configured, according to conventional means, to retrieveelectronic image data from the CCD image array. Alternatively, forexample, if the light intensity capture device included a photographicfilm, the image capture control unit could include a conventional imagescanning function configured to scan the captured image from thephotographic film, and thereby retrieve the electronic image data. Theinvention also includes other conventional methods of capturingelectronic image data, known to those of skill in this field.

The capture control unit 1104 controls the functions of the CCD 1102,and may also include and provide control for conventional photographicmechanical assemblies such as a shutter and/or a controllable aperture(not shown) to control aspects of capturing the image on the CCD 1102.Alternatively, one of skill in the image capture field will understandthat such mechanical assemblies controlled by the capture control unit1104 may be arranged in any convenient location along the light pathbetween the object and the image capture assembly, or between the lightsource and the image capture assembly.

According to the present embodiment, light is spread by the pattern inthe mask 304 which includes PSFs h_(n) (Equation 3) having diskfunctions p₁(x,y), p₂(x,y) and p₃(x,y) centered at points (x₁,y₁),(x₂,y₂) and (x₃,y₃), respectively, in the image capture space. Thus, animage capture device may be configured to include three distinct regionswithin a single light intensity capture device to receive three distinctpartial images produced by mask 304, such as the light intensity capturedevice 1102 in FIG. 11A. Further, the image capture device may includethree separate light intensity capture devices to receive the threedistinct partial images produced by mask 304, such as light intensitycapture devices 1106, 1108 and 1110 in the embodiment of image captureassembly 1112 shown in FIG. 11B.

FIG. 12A is a view of an example of a light intensity capture device1200 that includes a charge coupled device 1202 having three distinctregions 1204, 1206 and 1208. A central location in each region 1210,1212 and 1214, respectively, corresponds to a center of each of thethree partial images produced by the optical assembly 110. Inparticular, the coordinates of the points 1210, 1212 and 1214 correspondto (x₁,y₁), (x₂,y₂) and (x₃,y₃), respectively, from Equation 3.

FIG. 12B is an example of a two-dimensional intensity image according toEquation 1A captured by image capture assembly 120, including threepartial images 1216, 1218 and 1220 produced by optical assembly 110.Such a two-dimensional intensity image is converted into athree-dimensional image of the object by the image capture assembly 120,as described below.

FIGS. 13A-13C show other examples of ways in which the distinct regionsof a light intensity capturing device may be arranged, where eachdistinct region includes a center 1300. One of skill in the art willunderstand that the light intensity capturing device may be divided intothree or more zones in any convenient manner. For example, if the lightintensity capturing device includes a randomly addressable CCD, theboundaries of the zones may be arranged along convenient addressregions. Alternatively, if the light intensity capturing device includesa photographic film, the boundaries of the zones may be arrangedaccording to the geometries that are convenient for the dimensions andaspect ratio of the film.

According to Equation 1A above, an image capture assembly 120 receivesand captures light having a light intensity distribution given byo(x,y). To extract the object geometric information from the capturedimage, the image capture assembly operates on the captured image (i.e.,intensity function o(x,y)) according to Equation 1A. Thus, the objectgeometric information of the object s(x′,y′,z′) is given by thefollowing equation:

$\begin{matrix}{{s\left( {x^{\prime},y^{\prime},z^{\prime}} \right)} = {{O_{F}\left( {x,y} \right)}*{{\exp \left\lbrack {\frac{- {\pi}}{{\lambda\Delta}(z)}\left( {x^{2} + y^{2}} \right)} \right\rbrack}.}}} & (7)\end{matrix}$

where O_(F)(x,y) is a linear combination of the intensity distributionsin the partial images as follows:

O _(F)(x,y)=o ₁(x,y)[exp(−iθ ₃)−exp(−iθ ₂)]+o ₂(x,y)[exp(−iθ ₁)−exp(−iθ₃)]+o ₃(x,y)[exp(−iθ ₂)−exp(−iθ ₁)]  (8).

The extraction of the geometric information may be performed usingmethods from the field of digital holography, for example as describedin I. Yamaguchi, and T. Zhang, “Phase-shifting digital holography,” Opt.Lett. 22, 1268-1269 (1997), incorporated herein by reference.

In addition, the capture control unit 1104 may include functions forcombining the electronic data for each of the three partial imagesaccording to Equation 8, for extracting the object geometric informationaccording to Equation 7 and for providing the resulting object geometricinformation in a desired format. Alternatively, those functions may beperformed in a general purpose computer configured to receive the imagedata from the image capture assembly.

The object geometric information may be extracted as surface data, whichmay be suitable for use in applications such as physical modeling (e.g.,to create a computer model of the object) or three-dimensionalfabrication (e.g., to create a physical three-dimensional copy of theobject) applications. In addition, the object geometric information maybe displayed graphically, for example using two-dimensionalrepresentations of three-dimensional objects (e.g., a two-dimensionalprojection such as isometric projection, or a two-dimensionalrepresentation of a three-dimensional object that may be animated torotate the object around one or more axes to better illustrate thethree-dimensional object), or using direct three-dimensionalrepresentation of three-dimensional objects (e.g., holographic displayor projection).

FIG. 14 is a block diagram of an example of a capture control unit 1400that includes an image data processor 1402 that combines the electronicimage data according to Equation 7 to produce the object geometricinformation, and an object data output device 1404 to output the objectgeometric information. The image data processor 1402 may be implementedusing a conventional processor and conventional data processingsoftware. The object data output device 1404 may include any of a numberof conventional devices configured to utilize three-dimensional objectgeometric data such as a visual holographic display, a virtual realityenvironment display, a three-dimensional object fabrication device(e.g., laser sintering fabrication device, a digitally controlled lathe,etc. . . . ), a simulation model, a two-dimensional animation of amoving three-dimensional object, etc. . . . . The invention alsoincludes a capture control unit (not shown) that is configured toinclude an interface to an external control device, such as a computer,which may replace image data processor 1402 and object data output 1404,to flexibly perform the image data processing functions in a separatedevice.

In the embodiments described above, three different mask patterns havingthree different transmission functions (e.g., functions H₁, H₂, and H₃of equation (4B or 4A)) are combined in a single mask, three partialimages resulting from the mask patterns are simultaneously captured, andthe three partial images are combined to obtain geometric objectinformation. However, if the image capture assembly of FIG. 11A is used,the resulting resolution of the captured image may be reduced if thepixel array size is not increased three times so that each of the threeimages are the same resolution so that the three partial images may becaptured on a single light intensity capture device. Alternatively, ifthe image capture assembly of FIG. 11B is used, or if a single sensorwith three times the area is used, the resulting cost of the captureapparatus is increased by the cost of two additional light intensitycapture devices or the larger format sensor.

Another embodiment that varies a mask over time may not cause thepossible resolution reduction or cost increase of the precedingembodiment. In particular, in this embodiment, a mask may be varied overtime, resulting in three different partial images that vary over time.The three different partial images may be captured by an image captureassembly configured to capture images over time, and the three partialimages may be combined to extract the geometric information of theobject.

FIG. 15 is a block diagram of an embodiment of an optical apparatus 1500that varies the mask over time. Optical apparatus 1500 is similar to theembodiment of the optical apparatus 400 in FIG. 4, however, the opticalapparatus 1500 includes a controllable incoherent correlator 1502 and acontrollable image capture assembly 1506 that are controlled by a timingcontroller 1508, and the optical apparatus 1500 is configured to capturethree-dimensional or geometric information of object 130 using at leastthree different images captured at different times.

The controllable incoherent correlator 1502 is similar to the incoherentcorrelator 300 of the embodiment shown in FIG. 4. However, thecontrollable incoherent correlator 1502 includes a controllable mask1504 having a mask that may be controlled by the timing controller, tocontrollably transform the amplitude and phase of light received fromthe object. One or more spatial light modulators (SLMs), as described inFIGS. 6F and 6G, may be used in such a controllable incoherentcorrelator.

Further, the controllable image capture assembly 1506 is similar to theimage capture assembly 120 in the embodiment shown in FIG. 4. However,the controllable image capture assembly 1506 is further configured to becontrolled to capture and retrieve electronic image data by the timingcontroller 1508.

FIG. 16 is a block diagram of controllable mask 1504 that includes aspatial light modulator 1600 under the control of a mask controller1602. The mask controller 1602 controls the mask controller 1602 totransform light according to complex transform functions H₁, H₂ and H₃of Equation 4B, at times t₁, t₂ and t₃, as synchronized by the timingcontroller 1508. In an alternative embodiment, the mask controller 1602may be eliminated and the spatial light modulator 1600 may be controlleddirectly by the timing controller 1508, or by another external devicenot shown (e.g., an external computer operated controller). Imagecapture assembly 1506, also under the control of timing controller 1508captures three partial images at times t₁, t₂ and t₃ and combines thepartial images to obtain geometric information for object 130, asdescribed previously.

FIG. 17 is a block diagram of another embodiment of an optical apparatusin which the mask is varied over time. In this embodiment, a maskcontroller 1700 controls a mechanical position of a multimask 1712. Themultimask 1712 includes three masks 1702, 1708 and 1710 corresponding tothe masks H₁, H₂ and H₃ according to Equation 4B. The mask controllermoves the multimask 1712 in directions 1704 to place a correspondingmask between optical transforming assemblies 302 and 306 at times t₁,t₂, and t₃, under the control of timing controller 1508. Image captureassembly 1506, also under the control of timing controller 1508 capturesthree partial images at times t₁, t₂ and t₃ and combines the partialimages to obtain geometric object information for object 130, asdescribed previously. The multimask 1712 may include masks in a lineararrangement as shown in FIG. 17, or may include masks arranged in aradial arrangement, or any other suitable arrangement.

The three partial images may also be produced and capturedsimultaneously using an arrangement including three different opticalassemblies having different masks and arranged to each receive a portionof the light received from the object.

FIG. 18 is a block diagram of an embodiment of an optical apparatus 1800having optical assemblies 1802, 1804 and 1806 that are configured toeach receive a portion of the received light from the object by anarrangement of partially transmissive and reflective mirrors 1808, 1810and 1812 (e.g., “partially-silvered” mirrors). An image capture assembly120, such as the embodiments shown in FIGS. 11A and 11B, captures andprocesses the received partial images as described above.

Although the embodiments are described using only transmissive opticalelements (e.g., refractive lenses and transmissive masks) one of skillin the art will understand that the invention also includes alternativeembodiments in which one or more of the optical elements may be replacewith a corresponding reflective optical element, as desired.

FIG. 19 is an embodiment of an optical apparatus 1900 that is similar tooptical apparatus 400 shown in FIG. 4. However, optical apparatus 1900includes a reflective mask 1902 that is configured to reflect unmaskedlight, instead of transmitting the unmasked light as in mask 304. A beamsplitter 1904 redirects the light reflected by mask 1902 to the secondtransforming optical assembly 306.

FIGS. 20A and 20B are block diagrams of embodiments of opticalapparatuses 2000 and 2008, respectively, in which the first transformingoptical assembly is implemented using a reflective optical assembly. InFIG. 20A, a beam splitter 2004 directs a light received from the object130 to a reflective optical assembly 2002, which transforms the receivedlight and reflects the transformed light towards mask 304 and secondtransforming optical assembly 306. In the present embodiment, lighttransmitted from the second transforming optical assembly is reflectedto the image capture assembly 120 by a mirror 2006.

Similarly, in FIG. 20B, a reflective optical assembly 2010 receives alight from object 130, transforms the received light and reflects thetransformed light to a beam splitter 2012 which directs the transformedlight to mask 304, and so on.

Other arrangements of mirrors or beam splitters to conveniently directlight are also included in the present invention.

In the optical apparatus embodiments described above, when theelectromagnetic radiation received from the object includes a widebandwidth, it is possible to capture frequency information in the imagecapture assembly. Thus, it is possible for the image capture assembly todetermine a corresponding electromagnetic radiation frequency orfrequencies for each portion of the object. For example, when a whitelight is received at the optical assembly from the object, the imagecapture assembly may determine the color of each portion of the objectfrom the image captured by the image capture assembly.

In addition, it may be possible to increase the resolution of thecaptured three-dimensional information by reducing the bandwidth of thereceived light. For example, the resolution of the capturedthree-dimensional information may be increased by limiting the bandwidthof the received light to those frequencies of light close to the colorred. Such an increase in resolution may be obtained by filteringreceived or transmitted light in the optical assembly to have a reducedbandwidth using conventional filters, or by irradiating the object witha reduced bandwidth light source, using methods known by those of skillin the art.

However, images captured using a reduced light bandwidth may not includea sufficient level of information regarding the various colors of thereceived light, and therefore may not allow for the image captureassembly to determine colors of the object to a sufficiently high levelof accuracy. Accordingly, other embodiments of the invention may includeplural channels each configured to receive light and capture imageswithin different portions of the electromagnetic spectrum, and them tocombine the separately captured images to produce full spectrumthree-dimensional information regarding the object.

FIG. 21A is a block diagram of optical apparatus 2100 that receiveslight from object 130. The received light is partitioned into threelight portions 2103, 2105 and 2107 by light partitioning devices 2102,2104 and 2106, respectively. The three light portions 2103, 2105 and2107 each include a subset of the bandwidth of the received light. Forexample, light portion 2103 may include only light frequencies near thecolor red, light portion 2105 may include only light frequencies nearthe color green and light portion 2107 may include only lightfrequencies near the color blue. The light partitioning devices 2102,2104 and 2106 may include any combination of dichroic mirrors, colorfilters, mirrors or other partially transmissive frequency filteringdevices known to those of skill in the art.

The light portions 2103, 2105 and 2107 are received by opticalassemblies 2108, 2110 and 2112, respectively, which each may beconfigured to transform the received light as described above. That is,each of the optical assemblies 2108, 2110 and 2112 may transform a lightportion of the received light as described above (e.g., using threepartial mask patterns or a time varying pattern), and transmit thetransformed light to an image capture assembly 1112 that includes aseparate light capture assembly for each of the three partial maskpatterns, or to an image capture assembly 1100 (not shown) that includesa single light capture assembly configured to capture different imagesover time or different partial images within different regions of theassembly.

In addition, the optical apparatus 2100 includes an image combiningapparatus 2113 configured to receive image data representing the imagescaptured at image capture assemblies 1100 and combine the image data toproduce combined broadband three-dimensional information regarding theobject. For example, the optical apparatus 2100 may be able to capturefull-color three-dimensional information with a higher resolution thanthe embodiments described above.

FIG. 21B is a block diagram of an optical apparatus 2126 that receiveslight from object 130 and separates the received light into three lightportions 2115, 2117 and 2119 by light partitioning devices 2114, 2116and 2118, respectively . . . . The three light portions 2115, 2117 and2119 each include the entire bandwidth of the received light. Forexample, if the received light includes a white light then each of thethree light portions 2115, 2117 and 2119 also includes a white light.The light partitioning devices 2114, 2116 and 2118 may include anycombination of polychromatic mirrors, beam splitters or widebandtransmissive devices known to those of skill in the art.

The light portions 2115, 2117 and 2119 are received by opticalassemblies 2120, 2122 and 2124, respectively, which each may beconfigured to transform the received light as described above. That is,each of the optical assemblies 2120, 2122 and 2124 may transform a lightportion of the received light as described above (e.g., using threepartial mask patterns or a time varying pattern).

Further, each of the optical assemblies 2120, 2122 and 2124 may beconfigured to selectively filter out some received light frequencies.For example, the optical assemblies may include conventional colorfilters (not shown) to filter out certain light colors. Further, themask in each optical assembly may include light transforming regionshaving predetermined attenuation of received light frequencies, asdescribed above with respect to FIG. 6A. That is, each transform region614 may be configured to apply different amounts of amplitude reductionover different frequencies of the received light spectrum.

Each of the optical assemblies 2120, 2122 and 2124 transmits thetransformed portion of received light to an image capture assembly 1112that includes a separate light capture assembly or region of an assemblyfor each of the three partial mask patterns, or to an image captureassembly 1100 (not shown) that includes a single light capture assemblyconfigured to capture different images over time or different partialimages within different regions of the light capture assembly . . . .

Although the embodiments of FIGS. 21A and 21B include received lightthat is separated into three portions, other embodiments in which lightis separated into other numbers of portions are also included.

An incoherent correlator may equivalently be implemented with alternateoptical apparatuses other than the lens/mask/lens arrangements describedabove. For example, by applying the well-known thin lens approximationfor lenses, the incoherent correlator may be implemented with a singleoptical transforming element and a single mask, with either the mask orthe optical transforming element arranged to first receive light fromthe object. In addition, the optical assembly 110 may be implementedusing a single diffractive optical element. The equations 1-5 abovetherefore also apply to embodiments having a single transforming opticalassembly and a mask, and embodiments having an optical assemblyimplemented using only a single diffractive optical element.

FIG. 22A is a block diagram of an example of an optical apparatus 2200that is similar to the optical apparatus 400 shown in FIG. 4. However,the optical apparatus 2200 does not require a second transformingoptical element. Instead light is received from the object 130 byoptical transforming element 2202, which transforms the received lightand transmits the transformed light. The transformed light is receivedby mask 2204 which selectively transmits a portion of the transformedlight. Image capture assembly 120 receives and captures an image of theselectively transmitted light and obtains geometric informationregarding object 130 from the captured image, as described above.

FIG. 22B shows an example of an optical apparatus 2206 that is similarto the optical apparatus 400 shown in FIG. 4. However, the opticalapparatus 2206 does not require a first transforming optical element.Instead light is received from the object 130 by mask 2208 whichselectively transmits a portion of the received light. The secondoptical transforming element 2210 receives the selectively transmittedlight, transforms the received light and transmits the transformedlight. Image capture assembly 120 receives and captures an image of thetransformed light and obtains geometric information regarding object 130from the captured image, as described above.

There may be a relatively high cost to manufacture optical assemblieshaving a conventional incoherent correlator structure. An alternativeembodiment of the present invention the optical assembly may beimplemented using a single diffractive optical element (DOE) in place ofthe incoherent correlator.

A single DOE may replace the incoherent correlator (e.g., incoherentcorrelator 300 including first and second transforming opticalassemblies 302/306 and mask 304 in the embodiment shown in FIG. 4)described above. A DOE that is equivalent to the incoherent correlatoris the product of the mask filter function and the transmissionfunctions of the first and second transforming optical assemblies,H_(DOE), defined as follows:

$\begin{matrix}{{H_{DOE}\left( {u,v} \right)} = {{\exp \left\lbrack {\frac{- {2\pi}}{\lambda \; f}\left( {u^{2} + v^{2}} \right)} \right\rbrack}{\int{\int{{h\left( {x,y,0} \right)}{\exp \left\lbrack {\frac{- {\pi}}{\lambda \; f}\left( {x^{2} + y^{2}} \right)} \right\rbrack}{\exp \left\lbrack {\frac{2\pi}{\lambda \; f}\left( {{xu} + {vy}} \right)} \right\rbrack}{x}{y}}}}}} & \left( {9A} \right)\end{matrix}$

where f is the focal length of the first and second transforming opticalassemblies included in the DOE, h(x,y,0) is given above in Equation 3,and other parameters are as described with respect to Equation 4.

Further, the focal lengths of the lenses are not required to be thesame. When the focal lengths are different, the H_(DOE), is defined asfollows:

$\begin{matrix}{{H_{DOE}\left( {u,v} \right)} = {{\exp \left\lbrack {\frac{- {{\pi}\left( {f_{1} + f_{2}} \right)}}{\lambda \; f_{1}\lambda_{2}}\left( {u^{2} + v^{2}} \right)} \right\rbrack}{\int{\int{{h\left( {x,y,0} \right)}{\exp \left\lbrack {\frac{- {\pi}}{\lambda \; f_{2}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}{\exp \left\lbrack {\frac{2\pi}{\lambda \; f_{2}}\left( {{xu} + {vy}} \right)} \right\rbrack}{x}{y}}}}}} & \left( {9B} \right)\end{matrix}$

Note that although Equations 9A and 9B do not include literal P functionterms, p_(o)(x,y) is part of h(x,y,0), and when the integral inEquations 9A and 9B are solved, the convolution with P(u,v) is obtained.

FIG. 22C is a block diagram of an example of an optical apparatus 2212that is similar to the optical apparatus 400 shown in FIG. 4. However,the optical apparatus 2212 does not require first and secondtransforming optical elements. Instead light is received from the object130 by mask 2214 which transmits light based on a complex transformationof the received light. Mask 2214 may be implemented using only a singlediffractive optical element, as described above. Image capture assembly120 receives and captures an image of the transformed light and obtainsgeometric information regarding object 130 from the captured image, asdescribed above.

FIG. 23 is a block diagram of an embodiment of optical apparatus 100 inwhich the optical element 110 includes a reflective type diffractiveoptical element, for example, such as the diffractive optical elementsshown in FIGS. 5F and 5G. In this embodiment, the optical apparatus 100receives a light from object 130 along a receiving optical axis 2304.The optical apparatus 100 transforms and reflects the received light toproduce a transmitted light transmitted back along optical axis 2302.The transmitted light is reflected by a beam splitter 2300 to an imagecapture assembly 120 located along a capturing optical axis 2302 of theoptical apparatus. In the present embodiment, capturing optical axis2302 is arranged at an angle of approximately 90 degrees from thereceiving optical axis 2304. However, other angles between the receivingand capturing optical axes are also included in the invention.

With only two optical axes, the current embodiment may advantageouslyreduce a size of the optical assembly 100, while exhibiting lesssensitivity to axial variations than in conventional holography systems.

An objective-side optical assembly, such as an objective lens, a zoomlens, a macro lens, a microscope, a telescope, a prism, a filter, amonochromatic filter, a dichroic filter, a complex objective lens, awide-angle lens, a camera, a pin-hole, a light slit, a mirror, or anyother optical assembly may be placed between the optical assembly andthe object to collimate, focus, invert or otherwise modify the lightfrom the object, prior to the light being received at the opticalassembly. Such an arrangement may advantageously allow light fromobjects or portions of objects to be received, when it would not bepossible or practical to receive that light without the inclusion of theobjective-side optical assembly.

Further, an objective-side optical assembly may include refractive ordiffractive optical elements configured to at least partially cancel anydisadvantageous wavelength dispersal effects that may be caused by theoptical apparatus 100, as described by Goodman, “Introduction to FourierOptics,” 3rd Ed., Roberts & Company Publishers, 2005, at p. 212,incorporated herein by reference.

FIG. 24A shows an alternative embodiment including the features of theembodiment in FIG. 1, as well as an objective-side optical assembly 2400that receives light from the object and transmits a received light tothe optical assembly 110. The objective-side optical assembly 2400 inthe present embodiment includes a magnifying refracting objective lensthat produces a magnified image of the object 130 centered on an imageplane 2402. Thus, the present embodiment may capture more detailedgeometric information regarding a magnified portion of the object.

The present invention may also operate in conjunction with an existingsensor-less camera, which is understood herein to be any camera fromwhich the existing digital light sensor (e.g., CMOS device or CCD) orlight sensitive capture medium (e.g., film and film transport mechanism)has been removed, or moved away from the image plane of the camera toallow an apparatus according to the present invention to be used withthe remaining optical and mechanical components of the camera. Forexample, film, film transport mechanisms and the rear cover of anexisting 35 mm film camera may be removed and replaced with an opticalassembly and image capture device according to the present invention,thereby making the existing camera capable of capturingthree-dimensional information. Such an arrangement advantageously allowsthe present invention to conveniently take advantage of and operate withexisting photographic lenses, shutter systems and aperture controlsystems of existing cameras.

FIG. 24B shows an example of an embodiment of an optical apparatus 2404including features similar to the optical apparatus 100 shown in FIG. 1.In addition, the optical apparatus 2404 is configured to operate with anexisting sensor-less camera 2406, which receives light from the object130 along an optical axis 140, and manipulates the light usingconventional camera features (e.g., lens, ground glass focusing screen,shutter and aperture of the existing camera) to produce an image of theobject centered at image plane 2408. The optical apparatus 2404 includesa chassis having mechanical and electrical attachment features suitablefor coupling the optical apparatus 2404 to a portion of the existingsensor-less camera 2406 near the image plane produced by the optics ofthe existing sensor-less camera 2406 (e.g., as a replaceable “3D back”of the camera). The optical assembly 110 receives light from the imageof the object at the image plane 2408 and transmits a transformed light,which may be received, captured and processed to extractthree-dimensional information of an object by the image capture assembly120, as described above.

The present invention may also operate in conjunction with an existingcamera. In particular, the optical assembly in the embodiment describedherein may be used in conjunction with a conventional digital or filmcamera to illuminate the image plane of the conventional camera with aFresnel hologram or partial Fresnel holograms of the observed object.The conventional camera may be used to capture an image of the hologramfringe patterns using the corresponding conventional means (e.g., lightsensitive film or digital sensor), and image data corresponding to thefringe patterns may be converted into three-dimensional data of theobject using a general purpose computer.

The invention is not limited to a single DOE that includes atransmission function based on a linear combination of threetransmission functions each having a Fourier transforms of a FZP. On theother hand, the invention also includes receiving a portion of the lightfrom the object at each of three DOEs, which produce three partialimages that are combined.

FIG. 25 shows a block diagram of an embodiment of an optical apparatus2500 including partially reflective and transmissive mirrors 1808, 1810and 1812 that direct a light received from object 130 to each of threediffractive optical elements 2502, 2504 and 2506, respectively, whicheach perform a transforming function including a Fourier transform of aFZP. The image capture assembly 2508 extracts three-dimensionalinformation from an image of the light transmitted by the diffractiveoptical elements, similar to the manner described above.

In addition, alternative embodiments of the optical assembly 110 mayconsist of a single SLM as shown in FIG. 6F or one or more SLMs as shownin FIG. 6G.

Further, the invention is not restricted only to using three maskpatterns to produce three partial hologram images that are combined. Theinvention also includes using an off-axis holographic method thatemploys a single off-axis hologram instead of three masks.

During reconstruction of an image from an off axis hologram each term isdiffracted toward a different direction and therefore a desired angularseparation can be achieved even from a single hologram, by takingadvantage of the fact that angular separation in diffraction theory isdirectly translated to a spatial frequency separation. Thischaracteristic may be exploited based on the idea that, when performinga convolution between functions f and g, it is equivalent to transform fand g to the frequency domain by a Fourier transform to obtain functionsF and G, obtain a product of F and G, and transform the product back byan inverse Fourier transform. Thus, an optical apparatus that shifts aspatial frequency spectrum of a received light may be advantageouslyused to create a Fresnel hologram.

An optical apparatus that produces an off-axis Fresnel Zone Pattern(OAFZP) in response to point input light source can be used to convolvea received light rather than the FZPs in the embodiments above, andconvolution using the OAFZP based assembly will allow for a convenientseparation of terms in the frequency domain.

FIG. 26 shows an example of an OAFZP 2600.

To synthesize the off-axis FZP we may introduce a linear phase term tothe equation for the on-axis FZPs described above, to result in thefollowing OAFZP transformation function

$\begin{matrix}{{h\left( {x,y,z} \right)} = {{p_{z}\left( {x,y} \right)}\left\{ {{\frac{1}{\sqrt{2}}{\exp \left\lbrack {\frac{{\pi}\left( {x^{2} + y^{2}} \right)}{2{{\lambda\Delta}(z)}} + \frac{{2\pi}\left( {{\alpha \; x} + {\beta \; y}} \right)}{\lambda}} \right\rbrack}} + {\frac{1}{\sqrt{2}}{\exp \left\lbrack {\frac{- {{\pi}\left( {x^{2} + y^{2}} \right)}}{2{{\lambda\Delta}(z)}} - \frac{{2\pi}\left( {{\alpha \; x} + {\beta \; y}} \right)}{\lambda}} \right\rbrack}}} \right\}}} & (10)\end{matrix}$

Further, it is not necessary to use a mask or a spatial light modulatorto create such an OAFZP producing optical assembly. Alternatively, anarrangement of at least two lenses each shifted away from an opticalaxis of an image plane and arranged so that their focal points are atdifferent distances from an image plane may be used to produce an OAFZP.

FIG. 27 is a block diagram of a portion of an optical apparatusincluding a composite mask 2720 having lenses 2714 and 2716. Light 2718is received and refracted by lenses 2714 and 2716 towards lens 2702having focal length f 2712. In this example, lenses 2714 and 2716 areconfigured to have different focal lengths. Spherical waves 2706 and2704 produced by lenses 2714 and 2716, respectively, interfere with eachother to produce OAFZP 2708 at image plane 2710.

Although the example of FIG. 27 includes convex and concave lenses, theinvention includes any combination or permutation of convex and/orconcave lenses. Further, the lenses may have different focal lengths,and be arranged in a same plane, as shown in FIG. 27, or alternatively,the lenses may have the same or different focal lengths and be arrangedin different planes.

FIGS. 28A-D show examples of composite masks 2800, 2802, 2804 and 2806.Further, although the lenses shown in the composite mask examplesdescribed above are round lenses that cover only a portion of thecomposite mask plane or planes, the invention also includes other shapedlenses (e.g., cylindrical lenses) covering a portion or the entirety ofthe composite mask plane or planes. In addition, the invention includesreplacing one or both of the lenses in the composite mask with acorresponding diffractive optical element, or with a FZP.

Therefore, a composite mask that produces a single off-axis FZP mayreplace the masks or diffractive optical elements based on Fouriertransforms of FZPs in any of the optical apparatuses described above.However, to achieve further separation of terms in the frequency domaina pattern of lines may be projected on the object, or an optical gratinghaving an appropriate spatial frequency may be placed between the objectand the image capture plane to add a pattern of lines to the image ofthe object.

FIG. 29 is a block diagram of an embodiment of an optical apparatus 2900that is configured to receive light from object 130 and extractthree-dimensional information about object 130 from the received light.Optical apparatus 2900 includes an objective optical assembly 2902 thatreceives light from object 130 along an optical axis 140. The objectiveoptical assembly 2902 produces an image of the object 130 at an imageplane 2908. A grating 2904, located on the image plane 2908, adds apattern of lines to the image of the object 130 which propagates tofirst transforming lens 302. Further, the optical apparatus 2900includes a composite mask 2910 which transforms the light received fromthe first transforming lens 302, and transmits a transformed light.Second transforming lens 306 receives the transformed light andtransmits a further transformed light, and image capture assembly 120captures an image of the light and extracts three-dimensionalinformation from the captured image, as described above.

FIG. 30 is a detailed view of an embodiment of grating 2904 thatincludes low transmissivity regions 3002 and high transmissivity regions3000. A width 3006 of the low transmissivity regions 3002, and a width3004 of the high transmissivity regions 3000 are selected so thatcontrasting dark and light areas are observable in the resulting imageat the image capture device 120. Gratings including variable widths ofhigh and low transmissive regions may also be used.

As discussed above, the pattern of lines may also be applied to thelight illuminating the object or to the light received from the object.For example, light illuminating the object may pass through a linedtransparency configured to produce shadow lines on the object.

FIG. 31A is a block diagram of an embodiment of an optical apparatus3100 that may be used with a lined transparency 3102 to obtainthree-dimensional information of an object 130. Light from light source150 is shadowed by lines on the lined transparency 3102 to produce linedillumination on object 130. The lined illumination reflects from theobject 130 and is received by the first transforming lens 302 in theoptical apparatus 3100. Further, the optical apparatus 3100 includes acomposite mask 2910 which transforms the light received from the firsttransforming lens 302, and transmits a transformed light. Secondtransforming lens 306 receives the transformed light and transmits afurther transformed light, and image capture assembly 120 captures animage of the light and extracts three-dimensional information from thecaptured image, as described above.

Alternatively, it is not necessary to use the grating 2904. Instead, thelight coming from the object may be split into two beams, each of whichis transferred by a different off-axis lens toward a different portionof the filter. The filter and the system beyond the plane of the filterare similar to corresponding portions of the previous embodimentsdescribed in FIGS. 27, 29 and 31A.

FIG. 31B is a block diagram of an embodiment of an optical apparatus3100 that includes two off-axis lenses 2901 and 2903 and is configuredto obtain three-dimensional information of an object 130. Opticalapparatus 3100 includes an objective optical assembly 2902 that receiveslight from object 130 along an optical axis 140. Beam splitter 2908 andmirror 2909 transmit portions of the received light to mirrors 2907 and2905, respectively. The light reflected from mirrors 2905 and 2907propagates to first transforming lenses 2901 and 2903, respectively.Further, the optical apparatus 3100 includes a composite mask 2910 whichtransforms the light received from the first transforming lenses 2901and 2903, and transmits a transformed light. Second transforming lens306 receives the transformed light and transmits a further transformedlight, and image capture assembly 120 captures an image of the light andextracts three-dimensional information from the captured image, asdescribed above.

One of skill in the art will understand that the optical apparatusesdescribed above are not limited to capturing only reflected sunlight,but may also determine the shape and distance of object portions that donot reflect light but instead emit a fluorescent light, a black bodyradiation, a chemiluminescent light or other light produced by theobject, or objects that reflect or scatter light from sources other thanthe sun. In addition, an optical apparatus, according to the presentembodiment, is not limited to capturing only the external shape anddistance of objects, but may also capture information regarding internalportions of an object that radiate (i.e., reflect or fluoresce) lightfrom an internal portion through a transparent or translucent surface ofthe object to the optical apparatus.

The present invention is also not limited to capturing geometricinformation regarding an object using a Cartesian coordinate system(e.g., x, y, z), but also includes capturing geometric information usingany other coordinate system that may fully describe the shape, size andlocation of the object, such as a three-dimensional polar coordinatesystem (e.g., (p, θ, r), an earth referenced coordinate system such asthe global coordinate system (e.g., latitude, longitude, elevation), acoordinate system incorporating an ellipsoid earth model referencesystem such as WGS-84, an earth centered earth fixed Cartesiancoordinate system (ECEF) (e.g., x, y, z), Universal Transverse Mercator(UTM), Military Grid Reference System (MGRS), or World GeographicReference System (GEOREF), etc. . . . . Further, although Cartesian typemeasurement terms such as “vertical,” “horizontal” and “range” are usedthroughout the present description, those terms are intended to alsoinclude corresponding measurement terms in other reference systems, butwhich are omitted from the description herein for reasons of clarity andbrevity.

The use of the apparatuses is not limited to the field ofthree-dimensional imaging, but also includes uses in patternrecognition, target acquisition, and object identification, etc. . . .performed in three-dimensional space, for example as described in Y. Liand J. Rosen, “Object recognition using three-dimensional opticalquasi-correlation,” JOSA A 19, 1755-1762 (2002), incorporated herein byreference.

Advantages of the present invention may make embodiments of theinvention suitable for three-dimensional imaging applications that areimpossible or impractical without the present invention. For example,the present invention may be applied to capturing three-dimensionalmovies/video/television images, performing three-dimensional objectrecognition for moving objects or stationary objects from a moving orstationary platform (e.g., military targeting applications, roboticsensing applications, autonomous aid to vision impaired users, etc. . .. ), autonomous navigation and safety functions (e.g., automaticallyguide an automobile to stay on a road and avoid collisions with movingand stationary objects), weather sensing (e.g., capturethree-dimensional information regarding clouds or air masses detectedwith radar, visible light, or infrared and/or ultraviolet light, etc. .. . ), security functions (e.g., monitor locations and identity objectsin a room, monitor identities and locations of people in a building,three-dimensional synthetic radar, etc. . . . ), and three-dimensionalenvironmental mapping for virtual reality simulation (e.g., createthree-dimensional model of tourist destination for virtual visit), orthree-dimensional models of environments that are difficult orimpossible to observe directly (e.g., internal body cavities,microscopic environments, hazardous environments, extraterrestrialenvironments, underground or sea environments, remote environments, etc.. . . ).

Although examples described above deal with optical components andvisible light, the present invention also applies to receiving otherforms of electromagnetic radiation from an object and determiningthree-dimensional information of the object based on the receivedelectromagnetic radiation, such as x-ray radiation, microwave radiation,radio frequency radiation, and ultraviolet and infrared light. Forexample, embodiments of the invention described above may be modified toreplace optical components (e.g., lenses, mirrors, diffractive opticalelements, SLMs) with corresponding x-ray components, such as are knownin the art and as described in i) U.S. Pat. No. 6,385,291 to Takami, ii)Pereira et al., “Lithium x-ray refractive lenses,” Proc. SPIE 4502, 173(2001), and iii) Beguiristain et al., “Compound x-ray refractive lensesmade of polyimide,” Proc. SPIE, vol. 4144, pp. 155-164, each of which isincorporated herein by reference.

Further, for example, the present invention may be applicable as areplacement for existing x-ray imaging systems (e.g., CT scanners). Asthe present approach does not require any moving parts, x-ray imagingdone using an embodiment of the present invention advantageously mayproduce a scan more reliably, with higher resolution, greater speed andless total radiation exposure to the patient.

Each of the embodiments described above may be modified to replaceoptical elements with equivalent x-ray elements known to those of skillin the art to produce three-dimensional information based on a receivedx-ray radiation from an object (i.e., a three-dimensional x-ray image).For example, the present invention may be applicable as a replacementfor existing electron microscope technology.

Further the invention also applies to other forms of propagating energywaves, such as sound waves and may be applied to produce three-dimensionobject information using passive or active sonar.

Coherent light, which propagates according to the paraxialapproximation, is described mathematically as a convolution between aninput aperture and a quadratic phase function with an appropriateparameter in a denominator of an exponent power indicating a propagationdistance of the wave from the input aperture. Thus, the complexamplitude (i.e., the electrical field) distribution O(x,y) on sometransversal plane, in a distance z from the input plane, may be given(in the Fresnel approximation) by

$\begin{matrix}{{O\left( {x,y} \right)} = {\int{\int{{S\left( {x^{\prime},y^{\prime}} \right)}\exp \left\{ {\frac{\pi}{\lambda \; z}\left\lbrack {\left( {x - x^{\prime}} \right)^{2} + \left( {y - y^{\prime}} \right)^{2}} \right\rbrack} \right\} {x^{\prime}}{y^{\prime}}}}}} & (11)\end{matrix}$

where S(x′,y′) is the complex amplitude on the input aperture at thetransverse plane z=0, λ is the wavelength of the propagating light and(x′,y′), (x,y) are the coordinates of the input and output planes,respectively. For 3D objects, contributions from the object points areaccumulated to the following expression,

$\begin{matrix}{{O_{z}\left( {x,y} \right)} = {\int{\int{\int{{S\left( {x^{\prime},y^{\prime},z^{\prime}} \right)}\exp \left\{ {\frac{\; \pi}{\lambda \left( {z - z^{\prime}} \right)}\left\lbrack {\left( {x - x^{\prime}} \right)^{2} + \left( {y - y^{\prime}} \right)^{2}} \right\rbrack} \right\} {x^{\prime}}{y^{\prime}}{z^{\prime}}}}}}} & (12)\end{matrix}$

where (x′,y′,z′) are the coordinates of the input space. In aconventional holography approach that produces a Fresnel hologram, thecomplex amplitude O_(z)(x,y) may be interfered with a reference beam andthe intensity of the resulting interference pattern is recorded on aphotographic plate or a digital camera. However, according to thepresent invention, a convolution similar to Equation 12 may be performeddifferently using incoherent light, because the Fresnel propagationdescribed in Equation 11 may be valid only for coherent illumination.

For a two-dimensional (2D) input intensity function s(x,y) and anintensity point spread function (PSF) |h(x,y)|², a correlator outputintensity (e.g., of a correlator such as shown in FIG. 3) distributionmay be given by the following convolution,

o(x,y)=s(x,y)*h(x,y)² =∫∫s(x′,y′)h(x−x′,y−y′)² dx′dy′  (13)

where the asterisk denotes a 2D convolution, h(x,y) is the amplitude PSFin the system, but under coherent illumination. h(x,y) is related to the2D inverse Fourier transform of the filter function H(u,v) at plane P₂,as the following,

$\begin{matrix}{{h\left( {x,y} \right)} = {\int{\int{{H\left( {u,v} \right)}{\exp \left\lbrack {\frac{\; 2\pi}{\lambda \; f_{2}}\left( {{xu} + {yv}} \right)} \right\rbrack}{u}{v}}}}} & (14)\end{matrix}$

where f₂ is the focal length of the second lens in the correlator shownin FIG. 3. To solve for 3D objects rather 2D, a response of theincoherent correlator to a 3D input function may be determined. Further,although the input function is three-dimensional, the output and theconvolution remain two-dimensional. In fact the correlator response fora 3D input is,

o(x,y)=∫s(x,y,z)*|h(x,y,z)|² dz=∫∫∫s(x′,y′,z′)|h(x−x′,y−y′,z′)|²dx′dy′dz′  (15)

To calculate the general 3D amplitude PSF h(x,y,z) of the system, aresponse to a single point located at some point (x,y,z) in the vicinityof the rear focal point of the correlator may be determined. Such acalculation produces the 3D PSF of the system which may be used tocalculate the system response to any possible 3D input. Since the systemis known as space invariant it is correct to calculate the systemresponse to a point on the optical axis at some point (0,0,−z), and togeneralize the response toward a general location at (x,y,z). The inputpoint is located a distance f₁+z from the lens 302 at the point 308(i.e., 0,0,−z), as shown in FIG. 3.

The Fresnel integrals in Equations 11 and 12 can be used to calculatethe light distribution because a single monochromatic point source is bydefinition a spatial coherent source. By substituting the representationof a single point source, represented by a delta function δ(0,0,−z),into Equation 12 as the input S(x,y,z), the result on the plane of thefirst lens 302 is a diverging quadratic phase function as follows,

$\begin{matrix}{{O_{L_{1}}\left( {x,y} \right)} = {{\int{\int{\int{{\delta \left( {x^{\prime},y^{\prime},{z^{\prime} + z}} \right)}{\exp \left\lbrack {\frac{\; \pi}{\lambda \left( {f_{1} - z^{\prime}} \right)}\left\{ {\left( {x - x^{\prime}} \right)^{2} + \left( {y - y^{\prime}} \right)^{2}} \right\}} \right\rbrack}{x^{\prime}}{y^{\prime}}{z^{\prime}}}}}} = {\exp \left\lbrack {\frac{\; \pi}{\lambda \left( {f_{1} + z} \right)}\left( {x^{2} + y^{2}} \right)} \right\rbrack}}} & (16)\end{matrix}$

where f₁ is the focal length of the first lens in the correlator shownin FIG. 3. This quadratic phase function is known as the paraxialapproximation of the spherical wave propagating in the z direction, andthe paraxial approximation of a concave spherical lens transparency.This spherical wave propagates through the incoherent correlator andbeyond the correlator the beam becomes a converging spherical wave. Itmay be shown that at the plane where the beam is focused one gets theFourier transform of the transparency function of the mask H(u,v). ThisFourier transform is scaled according to the specific location of thefocal plane and is multiplied by a quadratic phase function.

Assuming that the three optical thin elements L₁, L₂ and H(u,v) of theincoherent correlator (e.g., elements 302, 306 and 304, respectively, inFIG. 3) are all located at the same plane, the diverging spherical waveand the two adjunct lenses L₁ and L₂ can be replaced by a singleequivalent lens having a focal length f_(e), as follows:

$\begin{matrix}{f_{e} = {\left( {\frac{1}{f_{1}} + \frac{1}{f_{2}} - \frac{1}{f_{1} + z}} \right)^{- 1} = \frac{f_{1}{f_{2}\left( {f_{1} + z} \right)}}{f_{1}^{2} + {z\left( {f_{1} + f_{2}} \right)}}}} & (17)\end{matrix}$

In a system having the equivalent lens in place of the correlator, oncethe system is illuminated by a plane wave, the complex amplitude on aback focal plane of equivalent lens L_(e) is related to the 2D Fouriertransform of the transparency function H(u,v). This means that thecomplex amplitude on the back focal plane, at a distance f, from theequivalent lens L_(e) is

$\begin{matrix}{{u\left( {x,y,z} \right)} = {A\; {\exp \left\lbrack {\frac{\; \pi}{\lambda \; {f_{e}(z)}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}{\int{\int{{H\left( {u,v} \right)}{\exp \left\lbrack {\frac{{- {2}}\; \pi}{\lambda \; {f_{e}(z)}}\left( {{xu} + {yv}} \right)} \right\rbrack}{u}{v}}}}}} & (18)\end{matrix}$

Note that the incoherent system is analyzed above according to the rulesof coherent diffraction theory because the beams are considered to havebeen emitted from a single infinitesimal point. Since the output of thesystem is located a distance f₂ from the equivalent lens L_(e), theoutput complex amplitude is obtained after a free propagation beyond theback focal plane of the equivalent lens L_(e).

Free propagation of coherent light may be obtained, as mentioned abovein Equation 11, as the result of convolution between the complexamplitude in the starting plane and a quadratic phase function.According to this, the output complex amplitude is,

$\begin{matrix}{{h\left( {x,y,z} \right)} = {{{u\left( {x,y,z} \right)}*{\exp \left\lbrack {\frac{\; \pi}{\lambda \left\lbrack {f_{2} - {f_{e}(z)}} \right\rbrack}\left( {x^{2} + y^{2}} \right)} \right\rbrack}} = {\left\{ {{\exp \left\lbrack {\frac{\; \pi}{\lambda \; {f_{e}(z)}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}{\int{\int{{H\left( {u,v} \right)}{\exp \left\lbrack {\frac{{- }\; 2\pi}{\lambda \; {f_{e}(z)}}\left( {{xu} + {yv}} \right)} \right\rbrack}{u}{v}}}}} \right\}*{\exp \left\lbrack {\frac{\; \pi}{\lambda \left\lbrack {f_{2} - {f_{e}(z)}} \right\rbrack}\left( {x^{2} + y^{2}} \right)} \right\rbrack}}}} & (19)\end{matrix}$

Note that although the function in Equation 19 deals with threedimensions, the convolution is always in 2D. Equation 19 expresses thegeneral 3D amplitude Point Spreading Function (PSF) of the system whenit is illuminated by coherent light. Further, Equation 19 can besimplified by writing explicitly the convolution integral, switching theorder of integration and using the well-known result of the Fouriertransform of quadratic phase function. Such a simplification reduces thefour integrals of Equation 19 to a double integral as follows:

$\begin{matrix}{{h\left( {x,y,z} \right)} = {{\exp \left\lbrack {\frac{\; \pi}{\lambda \; f_{2}}\left( {x^{2} + y^{2}} \right)} \right\rbrack} \times {\int{\int{{H\left( {u,v} \right)}{\exp \left\lbrack {\frac{{- }\; {\pi \left\lbrack {f_{2} - {f_{e}(z)}} \right\rbrack}}{\lambda \; f_{2}{f_{e}(z)}}\left( {u^{2} + v^{2}} \right)} \right\rbrack}{\exp \left\lbrack {\frac{{- }\; 2\pi}{\lambda \; f_{2}}\left( {{xu} + {yv}} \right)} \right\rbrack}{u}{z}}}}}} & (20)\end{matrix}$

Another equation used to synthesize the filter in the system is theexpression of the amplitude PSF for any point at the plane z=0, given bysubstituting f_(e)(0)=f₂ in Equation 20, as follows,

$\begin{matrix}{{h\left( {x,y,0} \right)} = {{\exp \left\lbrack {\frac{\; \pi}{\lambda \; f_{2}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}{\int{\int{{H\left( {u,v} \right)}{\exp \left\lbrack {\frac{{- }\; 2\pi}{\lambda \; f_{2}}\left( {{xu} + {yv}} \right)} \right\rbrack}{u}{{v}.}}}}}} & (21)\end{matrix}$

As described above, the intensity PSF for incoherent systems and forintensity distributions on the input and output planes is |h(x,y,z)|².The intensity PSF represents the impulse response of general incoherentsystems. By taking the absolute square of Equation 20 one finds that the3D intensity PSF is,

$\begin{matrix}{{{h\left( {x,y,z} \right)}}^{2} = {\begin{matrix}{\int{\int{{H\left( {u,v} \right)}{\exp \left\lbrack {\frac{{- }\; {\pi \left\lbrack {f_{2} - {f_{e}(z)}} \right\rbrack}}{\lambda \; f_{2}{f_{e}(z)}}\left( {u^{2} + v^{2}} \right)} \right\rbrack}}}} \\{{\exp \left\lbrack {\frac{{- }\; 2\pi}{\lambda \; f_{2}}\left( {{xu} + {yv}} \right)} \right\rbrack}{u}{v}}\end{matrix}}^{2}} & (22)\end{matrix}$

The general expression of Equation 22 can be used to compute the PSF fora given filter or the required filter for a given PSF.

According to Equation 17 the expression in the exponent of Equation 22is,

$\begin{matrix}{\frac{\left\lbrack {f_{2} - {f_{e}(z)}} \right\rbrack}{f_{2}{f_{e}(z)}} = {\frac{f_{2} - \frac{f_{1}{f_{2}\left( {f_{1} + z} \right)}}{f_{1}^{2} + {z\left( {f_{1} + f_{2}} \right)}}}{\frac{f_{1}{f_{2}^{2}\left( {f_{1} + z} \right)}}{f_{1}^{2} + {z\left( {f_{1} + f_{2}} \right)}}} = \frac{z}{f_{1}\left( {f_{1} + z} \right)}}} & (23)\end{matrix}$

Substituting Equation 23 into Equation 12 yields

$\begin{matrix}{{{h\left( {x,y,z} \right)}}^{2} = {\begin{matrix}{\int{\int{{H\left( {u,v} \right)}{\exp \left\lbrack {\frac{{- }\; \pi \; {z\left\lbrack {f_{2} - {f_{e}(z)}} \right\rbrack}}{\lambda \; {f_{1}\left( {f_{1} + z} \right)}}\left( {u^{2} + v^{2}} \right)} \right\rbrack}}}} \\{{\exp \left\lbrack {\frac{{- }\; 2\pi}{\lambda \; f_{2}}\left( {{xu} + {yv}} \right)} \right\rbrack}{u}{v}}\end{matrix}}^{2}} & (24)\end{matrix}$

The general expression of Equation 24 can be used to compute the PSF fora given filter or the required filter for a given PSF.

To obtain a Fresnel hologram, which is a convolution between any objectand a quadratic phase function, an incoherent intensity PSF in a shapeof a quadratic phase function with a number of cycles (Fresnel number)dependent on the distance z is selected. This may not be achieveddirectly because |h(x,y,z)|² is a positive real function while aquadratic phase function has negative and imaginary values.

One method of selecting such a PSF is to compose the PSF |h(x,y,z)|² asa sum of three terms, one of them is the required quadratic phasefunction, and their sum maintains the condition that |h(x,y,z)|² is apositive real function. Thus, a PSF such as shown in Equation 25

$\begin{matrix}{{{h\left( {x,y,z} \right)}}^{2} = {{p_{z}\left( {x,y} \right)}\left\{ {1 + {\frac{1}{2}{\exp \left\lbrack {\frac{\; \pi}{\lambda \; \Delta \; (z)}\left( {x^{2} + y^{2}} \right)} \right\rbrack}} + {\frac{1}{2}{\exp \left\lbrack {\frac{{- }\; \pi}{\lambda \; {\Delta (z)}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}}} \right\}}} & (25)\end{matrix}$

satisfies this condition, where Δ(z) is a parameter linearly related tothe distance z and p_(z)(x,y) is a disk function with the diameter d(z),different for different values of z, that indicates the limitingaperture of a corresponding Fresnel Zone Pattern (FZP). The amplitudePSF for this choice is

$\begin{matrix}\begin{matrix}{{h\left( {x,y,z} \right)} = \sqrt{{p\left( {x,y} \right)}\left\{ {1 + {\cos \left\lbrack {\frac{\; \pi}{\lambda \; {\Delta (z)}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}} \right\}}} \\{= {\sqrt{2}{p\left( {x,y} \right)}{\cos \left\lbrack {\frac{\; \pi}{2\lambda \; {\Delta (z)}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}}} \\{= {{p_{z}\left( {x,y} \right)}\left\{ {{\frac{1}{\sqrt{2}}{\exp \left\lbrack \frac{{\pi}\left( {x^{2} + y^{2}} \right)}{2\lambda \; {\Delta (z)}} \right\rbrack}} + {\frac{1}{\sqrt{2}}{\exp \left\lbrack \frac{- {{\pi}\left( {x^{2} + y^{2}} \right)}}{2\lambda \; \Delta \; (z)} \right\rbrack}}} \right\}}}\end{matrix} & (26)\end{matrix}$

Note that a possible arbitrary pure phase term can multiply h(x,y,z)without affecting the square magnitude of h(x,y,z) given in Equation 25.However in order to get a Fresnel hologram of all the object's points,it is preferred that h(x,y,z) remains as a sum of two quadratic phaseterms along the propagation axis. Of the possible phase functions thatcan multiply h(x,y,z), only a quadratic phase function may satisfy thecondition that h(x,y,z) is a sum of two quadratic phase terms afterpropagating a distance. Accordingly, it is appropriate to assume thath(x,y,z) is a sum of two quadratic waves with the same magnitude ofFresnel number but with opposite signs, as given in Equation 26.Further, as described below, two quadratic waves with different Fresnelnumbers may be used in an optimized solution.

Based on the desired h(x,y,z), H(u,v) may be calculated by inversingEquation 21, to produce the following filter function in Equation 27.

$\begin{matrix}{{H\left( {u,v} \right)} = {\int{\int{{h\left( {x,y,0} \right)}{\exp \left\lbrack {\frac{{- }\; \pi}{\lambda \; f_{2}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}{\exp \left\lbrack {\frac{\; 2\pi}{\lambda \; f_{2}}\left( {{xu} + {yv}} \right)} \right\rbrack}{x}{y}}}}} & (27)\end{matrix}$

Substituting Equation 26 into Equation 27 yields Equation 28

$\begin{matrix}{{H\left( {u,v} \right)} = {\left\{ {{\frac{1}{2}{\exp \left\lbrack {\frac{\; \pi}{\lambda \; \gamma_{1}}\left( {u^{2} + v^{2}} \right)} \right\rbrack}} + {\frac{1}{2}{\exp \left\lbrack {\frac{\; \pi}{\lambda \; \gamma_{2}}\left( {u^{2} + v^{2}} \right)} \right\rbrack}}} \right\}*{P\left( {u,v} \right)}}} & (28)\end{matrix}$

where P(u,v) is the Fourier transform of p_(o)(x,y). Note that H(u,v) is2D function which determines the dependency of h(x,y,z) along thetransverse coordinates (x,y). The dependency of h(x,y,z) along the zaxis is dictated by the location of input source point.

The intensity PSF may be obtained by substituting the filter function ofEquation 28 into Equation 22. Substituting Equation 17 into the exponentexpression of Equation 22 yields,

$\begin{matrix}{\frac{\left\lbrack {f_{2} - {f_{e}(z)}} \right\rbrack}{f_{2}{f_{e}(z)}} = {\frac{f_{2} - \frac{f_{1}{f_{2}\left( {f_{1} + z} \right)}}{f_{1}^{2} + {z\left( {f_{1} + f_{2}} \right)}}}{\frac{f_{1}{f_{2}^{2}\left( {f_{1} + z} \right)}}{f_{1}^{2} + {z\left( {f_{1} + f_{2}} \right)}}} = \frac{z}{f_{1}\left( {f_{1} + z} \right)}}} & (29)\end{matrix}$

Assuming that the filter function is

$\begin{matrix}{{H\left( {u,v} \right)} = {\left\{ {{\frac{1}{2}{\exp \left\lbrack {\frac{\; \pi}{\lambda \; \gamma}\left( {u^{2} + v^{2}} \right)} \right\rbrack}} + {\frac{1}{2}{\exp \left\lbrack {\frac{{- }\; \pi}{\lambda \; \gamma}\left( {u^{2} + v^{2}} \right)} \right\rbrack}}} \right\}*{P_{o}\left( {u,v} \right)}}} & (30)\end{matrix}$

therefore, Equation 22 becomes,

$\begin{matrix}{{{h\left( {x,y,z} \right)}}^{2} = {{\int{\int{\left\{ {{\frac{1}{2}{\exp \left\lbrack {\frac{\; \pi}{\lambda \; \gamma}\left( {u^{2} + v^{2}} \right)} \right\rbrack}} + {\frac{1}{2}{\exp \left\lbrack {\frac{{- }\; \pi}{\lambda \; \gamma}\left( {u^{2} + v^{2}} \right)} \right\rbrack}}} \right\}*{P_{o}\left( {u,v} \right)} \times {\quad{{\exp \left\lbrack {\frac{{- }\; \pi}{\lambda}\frac{z}{f_{1}\left( {f_{1} + z} \right)}\left( {u^{2} + v^{2}} \right)} \right\rbrack}{\exp \left\lbrack {\frac{{- }\; 2\; \pi}{\lambda \; f_{2}}\left( {{xu} + {yv}} \right)} \right\rbrack}{u}\; {v}}}^{2}}}}}} & (31)\end{matrix}$

After summation corresponding terms, the result is,

$\begin{matrix}{{{h\left( {x,y,z} \right)}}^{2} = {{\int{\int\left\{ {{\frac{1}{2}{\exp \left\lbrack \frac{\; {\pi \left( {f_{1}^{2} + {f_{1}\; z} - {\gamma \; z}} \right)}\left( {u^{2} + v^{2}} \right)}{\lambda \; \gamma \; {f_{1}\left( {f_{1} + z} \right)}} \right\rbrack}} + {\quad{\left. \quad{\frac{1}{2}{\exp \left\lbrack \frac{{- }\; {\pi \left( {f_{1}^{2} + {f_{1}z} + {\gamma \; z}} \right)}\left( {u^{2} + v^{2}} \right)}{\lambda \; \gamma \; {f_{1}\left( {f_{1} + z} \right)}} \right\rbrack}} \right\}*{\quad{{P_{o}\left( {u,v} \right)}{\exp \left\lbrack {\frac{{- }\; 2\; \pi}{\lambda \; \gamma_{2}}\left( {{xu} + {yv}} \right)} \right\rbrack}{u}\; {v}}}^{2}}}} \right.}}}} & (32)\end{matrix}$

Calculating the Fourier transform,

$\begin{matrix}{{{h\left( {x,y,z} \right)}}^{2} = {\left\{ {{\frac{1}{2}{\exp \left\lbrack \frac{{- }\; \pi \; \gamma \; {f_{1}\left( {f_{1} + z} \right)}\left( {x^{2} + y^{2}} \right)}{\lambda \; {f_{2}^{2}\left( {{f_{1}^{2}\gamma} + {f_{1}z} - {\gamma \; z}} \right)}} \right\rbrack}} + {\quad{\left. \quad{\frac{1}{2}{\exp \left\lbrack \frac{\; \pi \; \gamma \; {f_{1}\left( {f_{1} + z} \right)}\left( {x^{2} + y^{2}} \right)}{\lambda \; {f_{2}^{2}\left( {{f_{1}^{2}\gamma} + {f_{1}z} - {\gamma \; z}} \right)}} \right\rbrack}} \right\} {p_{o}\left( {x,y} \right)}}}^{2}} \right.}} & (33)\end{matrix}$

Calculating the square magnitude yields,

$\begin{matrix}{{{h\left( {x,y,z} \right)}}^{2} = {1 + {\frac{1}{4}{\exp \left\lbrack \frac{\; 2\; \pi \; \gamma \; {f_{1}^{2}\left( {f_{1} + z} \right)}^{2}\left( {x^{2} + y^{2}} \right)}{\lambda \; {f_{2}^{2}\left( {{f_{1}^{2}\left( {f_{1} + z} \right)}^{2} - {\gamma^{2}z^{2}}} \right)}} \right\rbrack}} + {\frac{1}{4}{\exp \left\lbrack \frac{{- }\; 2\; \pi \; \gamma \; {f_{1}^{2}\left( {f_{1} + z} \right)}^{2}\left( {x^{2} + y^{2}} \right)}{\lambda \; {f_{2}^{2}\left( {{f_{1}^{2}\left( {f_{1} + z} \right)}^{2} - {\gamma^{2}z^{2}}} \right)}} \right\rbrack}}}} & (34)\end{matrix}$

The parameter Δ(z) is,

$\begin{matrix}{{\Delta (z)} = \frac{\left( {{f_{1}^{2}\left( {f_{1} + z} \right)}^{2} - {\gamma^{2}z^{2}}} \right)f_{2}^{2}}{2\; \gamma \; {f_{1}^{2}\left( {f_{1} + z} \right)}^{2}}} & (35)\end{matrix}$

Equation 35 gives the value of Δ(z), the distance of a reconstructedimage point as a function of the object point's location on the z axis,for the general choice of y_(1,2)=±y.

The derivative of Δ(z) yields the axial magnification as a function ofz. The derivative of Δ(z) given in Equation 35 is,

$\begin{matrix}{\frac{{\Delta (z)}}{z}\frac{{- z}\; \gamma \; f_{2}^{2}}{{f_{1}\left( {f_{1} + z} \right)}^{3}}} & (36)\end{matrix}$

Equation 36 indicates that there is a point at z=0, which is at thefront focal point in which the axial magnification is zero. This pointis also an extreme point of the function Δ(z). This means that for anobject positioned at this location, points on one side (e.g., objectpoints where z<0) yield the same hologram as other points from the otherside (e.g., object points where z>0). The result might be areconstruction of an axially folded image. Therefore, recording ahologram of an object located at this point may be advantageouslyavoided.

The immediate conclusion from this curve is that one cannot record ahologram of an object that has points from both sides of the forbiddenpoint. This is because every two points to the left and to the right ofz=0 induce the same hologram which is actually a FZP with the same valueof Δ(z). In other words, from the recorded hologram one cannot knowwhether the object is located at z or at −z. A solution to this problemmay be to record holograms of objects that are all located only at oneside of the point z=0.

We also see from Equation 36 that Δ(z) is not a linear function of z.This phenomenon might introduce distortion of the image if the objecthas considerable depth. However, this depth distortion can becompensated during computer reconstruction of the three-dimensionalimage based on the known curve of Δ(z).

Equation 36 also shows that that in the region z>0 there is a pointwhere dΔ(z)/dz gets its maximum value. This point may be convenient forlocating the object because in the vicinity of this point themagnification may be maximal and approximately linear. This point may befound by comparing a second derivative of Δ(z) to zero. The optimalpoint is at z_(o)=f₁/2. Substituting the value of z_(o) back intoEquation 36 yields the following axial magnification

$\begin{matrix}{M_{A} = {\frac{{\Delta (z)}}{z} = \frac{{- 4}\; \gamma \; f_{2}^{2}}{27\; f_{1}^{3}}}} & (37)\end{matrix}$

Thus, Equation 37 provides a basis for selecting the value of y. In anyhologram without distortions during the reconstruction the axial and thetransverse magnifications are equal. Therefore, the transversemagnification of the inventive system is M_(T)=−f₂/f₁. Substituting thenon-distortion constraint that M_(T)=M_(A) into Equation 37 yields thefilter parameter as follows,

$\begin{matrix}{\gamma = \frac{27\; f_{1}^{2}}{4\; f_{2}}} & (38)\end{matrix}$

For example, when magnifications of the two lenses are −0.5 (i.e.,f₁=2f₂) the filter parameter y is 27f₂. Thus, in the embodiment of FIG.3, for the simpler case of f₁=2f₂, one quadratic phase has a focallength resulting in y=27f. For symmetry, the other quadratic phase maylikewise be selected to have a focal length resulting in y₂=−27f₂.Substituting these values into Equation 30 yield the following filterfunction:

$\begin{matrix}{{H\left( {u,v} \right)} = {\left\{ {{\frac{1}{2}{\exp \left\lbrack {\frac{\; \pi}{27\; \lambda \; f_{2}}\left( {u^{2} + v^{2}} \right)} \right\rbrack}} + {\frac{1}{2}{\exp \left\lbrack {\frac{{- }\; \pi}{27\; \lambda \; f_{2}}\left( {u^{2} + v^{2}} \right)} \right\rbrack}}} \right\}*{P_{o}\left( {u,v} \right)}}} & (39)\end{matrix}$

Note, that such a symmetric hologram has real values only, and thisproperty may be used to advantageously implement a mask. Substitutingthe filter function of Equation 39 into Equation 22 yields intensity PSFof the form of Equation 25, where Δ(z) is given by Equation 35 asfollows,

$\begin{matrix}\begin{matrix}{{\Delta (z)} = \frac{\left\lbrack {{f_{1}^{2}\left( {f_{1} + z} \right)}^{2} - {\frac{27^{2}}{4}f_{1}^{2}z^{2}}} \right\rbrack \frac{f_{1}^{2}}{4}}{2\frac{27}{2}{f_{1}^{3}\left( {f_{1} + z} \right)}^{2}}} \\{= \frac{\left\lbrack {{4\left( {f_{1} + z} \right)^{2}} - {27^{2}z^{2}}} \right\rbrack f_{1}}{108\left( {f_{1} + z} \right)^{2}}}\end{matrix} & (40)\end{matrix}$

In the point z_(o)=f₁/2 of maximum, and almost linear, magnification,Δ(z_(o)) is

$\begin{matrix}{{\Delta \left( z_{o} \right)} = {{- \frac{693}{972}}f_{1}}} & (41)\end{matrix}$

Note that when the hologram is reconstructed, twin images are obtainedalong the z axis at the vicinity of the points ±Δ(z) from the hologramplane. Solving the twin image reconstruction problem is furtherdescribed below.

Following is an example of how parameters may be selected forfabrication of a mask according to the present invention. Assuming thatan SLM used as the filter medium has N×N pixels in a rectangle area ofsize D×D, a FZP with ±y parameters may be displayed on the SLM, wherethe width of a thinnest possible ring is given by δ=|y|λ/D and δ=D/N.Therefore, from the equation D/N=|y|λ/D one gets |y|=D²/Nλ. For an SLMhaving D≈2 cm, N≈1000 pixels, the result is |y|≈80 cm in the visiblelight regime where λ=0.5 μm. According to Equation 22, and thediscussion after that, y_(1,2)=±27f₂ and therefore f₂=6 cm and f₁≈f₂/2=3cm.

Equation 25 describes the intensity PSF captured by an image capturedevice according to the present invention. This PSF has three additiveterms that are all concentrated in the center of the image captureplane. Therefore, convolution of the object function with such anintensity PSF yields three overlapped non-separated terms. However, itis desired to extract only a desired convolution term between the objectand a single quadratic phase function among the three convolutions withthree terms of the intensity PSF. A desired convolution between theobject and and a single quadratic phase function among the threeconvolutions with three terms of Equation 22 may be extracted usingmethods similar to those in digital holography, for example as describedby I. Yamaguchi, and T. Zhang, “Phase-shifting digital holography,” Opt.Lett. 22, 1268-1269 (1997), which is incorporated herein by reference.

The correlator may perform three operations of convolution between theobject and three PSFs equipped with three different constant phasevalues. These PSFs may be synthesized by introducing three filter maskswith three different constant phase values as follows,

$\begin{matrix}{{{H_{n}\left( {u,v} \right)} = {\left\{ {{\frac{1}{2}{\exp \left\lbrack {{\frac{\; \pi}{\lambda \; \gamma}\left( {u^{2} + v^{2}} \right)} + \frac{\; \theta_{n}}{2}} \right\rbrack}} + {\frac{1}{2}{\exp \left\lbrack {{\frac{\; \pi}{\lambda \; \gamma}\left( {u^{2} + v^{2}} \right)} - \frac{\; \theta_{n}}{2}} \right\rbrack}}} \right\}*{P\left( {u,v} \right)}}},{n = 1},2,3} & (42)\end{matrix}$

By the relation of Equation 24, it can be shown that the three filtersinduce three intensity PSFs as follows,

$\begin{matrix}{{{h_{n}\left( {x,y,z} \right)}}^{2} = {{p_{z}\left( {x,y} \right)} \times \left\{ {1 + {\frac{1}{2}{\exp \left\lbrack {{\frac{\; \pi}{\lambda \; {\Delta (z)}}\left( {x^{2} + y^{2}} \right)} + {\; \theta_{n}}} \right\rbrack}} + {\frac{1}{2}{\exp \left\lbrack {{\frac{- {\pi}}{\lambda \; {\Delta (z)}}\left( {x^{2} + y^{2}} \right)} - {\; \theta_{n}}} \right\rbrack}}} \right\}}} & (43)\end{matrix}$

Substituting the three PSFs of Equation 43 into Equation 15 yields theoutput intensity images that may be recorded by a camera or othersuitable image capture device (e.g., CCD, CMOS, photographic film, etc.. . . ):

$\begin{matrix}{{{o_{n}\left( {x,y} \right)} = {{\int{{s\left( {x,y,z} \right)}*{{h_{n}\left( {x,y,z} \right)}}^{2}{z}}} = {{\int{\left\lbrack {{s\left( {x,y,z} \right)}*{p_{z}\left( {x,y} \right)}} \right\rbrack {z}}} + {\frac{1}{2}{\exp \left( {\; \theta_{n}} \right)}{\int{{s\left( {x,y,z} \right)}*{p_{z}\left( {x,y} \right)}{\exp \left\lbrack {\frac{\; \pi}{\lambda \; {\Delta (z)}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}{z}}}} + {\frac{1}{2}{\exp \left( {{- }\; \theta_{n}} \right)}{\int{{s\left( {x,y,z} \right)}*{p_{z}\left( {x,y} \right)}{\exp \left\lbrack {\frac{{- }\; \pi}{\lambda \; {\Delta (z)}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}{z}}}}}}},{n = 1},2,3} & (42)\end{matrix}$

From these three images, a single term of convolution between the objects(x,y) and one of the quadratic phases may be extracted. A possibleformula to isolate such a single convolution is

O _(F)(x,y)=o ₁(x,y)[exp(−iθ ₃)−exp(−iθ ₂)]+o ₂(x,y)[exp(−iθ ₁)−exp(−iθ₃)]+o ₃(x,y)[exp(−iθ ₂)−exp(−iθ ₁)]  (43)

O_(F)(x,y) is a final complex valued hologram which satisfies therelation,

$\begin{matrix}{{O_{F}\left( {x,y} \right)} = {\int{{s\left( {x,y,z} \right)}*{p\left( {x,y} \right)}{\exp \left\lbrack {\frac{\; \pi}{\lambda \; {\Delta (z)}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}{z}}}} & (44)\end{matrix}$

The function O_(F)(x,y) is the final hologram which contains informationon the one 3D image only. Such an image s(x,y,z) can be reconstructedfrom O_(F)(x,y) by calculating the inverse operation to Equation 45, asfollows,

$\begin{matrix}{{s\left( {x,{y;z}} \right)} = {{O_{F}\left( {x,y} \right)}*{\exp \left\lbrack {\frac{{- }\; \pi}{\lambda \; {\Delta (z)}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}}} & (45)\end{matrix}$

Subsequently, the process of obtaining a single hologram with goodseparation between the three terms will be described. However, practicalaspects of performing the convolution with three different PSFs aredescribed first. There are several ways in which the filters may bemultiplexed to produce the three partial images. For example, a timemultiplexing system, such as the embodiment shown in FIG. 15,multiplexes the filters over time. Alternatively, the multiplexing maybe done in the output plane of a single channel, for example as in theembodiment shown in FIG. 4. When a single point source is introduced atthe point (0,0,0) the system's PSF is a pattern of 3 FZP with 3different phases, distributed at 3 separated locations on the outputplane. This 2D amplitude PSF is given by,

$\begin{matrix}{{{h\left( {x,y,0} \right)} = {\sum\limits_{n = 1}^{3}\; {\left( {{\frac{1}{\sqrt{2}}\exp \left\{ {{\frac{\; \pi}{2\lambda \; {\Delta (0)}}\left\lbrack {\left( {x - x_{n}} \right)^{2} + \left( {y - y_{n}} \right)^{2}} \right\rbrack} + \frac{\; \theta_{n}}{2}} \right\}} + {\frac{1}{\sqrt{2}}\exp \left\{ {{\frac{{- }\; \pi}{2\lambda \; {\Delta (0)}}\left\lbrack {\left( {x - x_{n}} \right)^{2} + \left( {y - y_{n}} \right)^{2}} \right\rbrack} - \frac{\; \theta_{n}}{2}} \right\}}} \right){p_{o}\left( {{x - x_{n}},{y - y_{n}}} \right)}}}},} & (46)\end{matrix}$

where (x_(n),y_(n)) is the center point of the nth FZP. h(x,y,0) ofEquation 46 may be used to synthesize the filter H(u,v) by Fouriertransform of h(x,y,0). For synthesizing the diffractive optical element(DOE) in a lensless system, for example the embodiment shown in FIGS. 1and 22C, one may multiply the filter function by the transmissionfunction of the two spherical lenses, to identify the overalltransmission function of the DOE as follows,

$\begin{matrix}{{{H\left( {u,v} \right)} = {{\exp \left\lbrack {\frac{{- }\; {\pi \left( {f_{1} + f_{2}} \right)}}{\lambda \; f_{1}f_{2}}\left( {u^{2} + v^{2}} \right)} \right\rbrack}{\int{\int{{h\left( {x,y,0} \right)}{\exp \left\lbrack {\frac{{- }\; \pi}{\lambda \; f_{2}}\left( {x^{2} + y^{2}} \right)} \right\rbrack}{\exp \left\lbrack {\frac{\; 2\; \pi}{\lambda \; f_{2}}\left( {{xu} + {vy}} \right)} \right\rbrack}{x}{y}}}}}},} & (47)\end{matrix}$

where h(x,y,0) is given in Equation 46.

A further embodiment includes a method of recording digital Fresnelholograms under incoherent illumination. According to this embodiment,the reflected white light from a 3-D object propagates through adiffractive optical element (DOE) and is recorded by a digital camera.Three holograms are recorded sequentially each with a different phasefactor of the DOE. The three holograms are superposed in the computersuch that the result is a complex valued Fresnel hologram. The 3-Dproperties of the object are revealed by reconstructing this hologram inthe computer. To the best of our knowledge, the demonstrated hologram isthe first digital hologram recorded without using laser light.

A system according to this embodiment is shown in FIG. 33. A white lightsource 3301 illuminates a 3-D object 3302 and the reflected light 3303from the object is captured by a CCD camera 3304 after passing through alens L 3305 and a DOE displayed on a spatial light modulator (SLM) 3306.The specific SLM in this experiment operates in reflection mode, but itis well understood that the same principles and analysis are valid fortransmission SLM as well. In general, such system can be analyzed as anincoherent correlator, where the DOE function is considered as thesystem's transfer function. However, in this work we find it easier toregard the system as an incoherent interferometer, where the gratingdisplayed on the SLM is considered as a beam splitter. As is common insuch cases, we analyze the system by following its response to an inputobject of a single infinitesimal point. Knowing the system's pointspread function (PSF), enables one to realize the system operation forany general object. Analysis of a beam originated from narrow bandinfinitesimal point source is done using Fresnel diffraction theory (J.Goodman, Introduction to Fourier Optics, 2^(nd) ed., McGraw-Hill, NewYork, 1996, pp. 63-95 (Chapter 4)) since such a source is coherent bydefinition.

A Fresnel hologram of a point object is obtained when the twointerfering beams are, for instance, plane and spherical beams. Such agoal is achieved if the DOE's reflection function R(x,y) is of the form,

$\begin{matrix}{{{R\left( {x_{D},y_{D}} \right)} = {{\frac{1}{2} + {\frac{1}{2}{\exp \left\lbrack {{{- \frac{\; \pi}{\lambda \; a}}\left( {x_{D}^{2} + y_{D}^{2}} \right)} + {\; \theta}} \right\rbrack}}} = {\frac{1}{2} + {\frac{1}{2}{Q\left( {- \frac{1}{a}} \right)}{\exp \left( {\; \theta} \right)}}}}},} & (48)\end{matrix}$

Where λ is the central wavelength, and for the sake of shortening, thequadratic phase function is designated by the function Q such thatQ(b)=exp[iπb/λ(x²+y²)]. The constant term of ½ in Eq. (48) contributesthe plane wave, and the quadratic phase term is responsible on thespherical wave. The angle θ plays an important rule later in thecomputation process in order to get rid of the twin image and the biasterm.

A point source located at the point (0,0,z_(s)) a distance f−z_(s) froma spherical positive lens, with f focal length, induces on the lensplane a diverging spherical wave of the form of Q(1/f−z_(s)). Rightafter the lens, which has a transmission function of Q(−1/f), thecomplex amplitude of the wave isQ(1/f−z_(s))Q(−1/f)=Q[z_(s)/f(f−z_(s))]. After propagating additionaldistance of d₁ till the DOE plane, the complex amplitude becomesQ{z_(s)/[f(f−z_(s))+z_(s)d₁]}. Right after the DOE, with the reflectionfunction given in Eq. (1), the complex amplitude is related toQ{z_(s)/[f(f−z_(s))+z_(s)d]}[1+Q(−1/a)exp(iθ)]. Finally, in the CCDplane a distance d₂ from the DOE, the intensity of the recorded hologramis,

$\begin{matrix}{{{I_{P}\left( {x,y} \right)} = {A{{{Q\left\lbrack \left( {\frac{f\left( {f - z} \right)}{z} + d_{1} + d_{2}} \right)^{- 1} \right\rbrack} + {{Q\left\lbrack \left( {\frac{{{af}\left( {f - z} \right)} + {azd}_{1}}{{za} - {f\left( {f - z} \right)} - {zd}_{1}} + d_{2}} \right)^{- 1} \right\rbrack}{\exp \left( {\; \theta} \right)}}}}^{2}}},} & (49)\end{matrix}$

where A is a constant. The first term of Eq. (49) is now approximated toa constant by assuming that z is much smaller than f Since the system isshift invariant the result of I_(P)(x,y), after calculating the squaremagnitude in Eq. (49), can be generalized to a PSF for any source pointlocated at any point (x_(s),y_(s),z_(s)), as follows,

$\begin{matrix}{{I_{P}\left( {x,y} \right)} = {A\left( {{2 + {\exp \left\{ {{\frac{\; \pi}{\lambda \; {\gamma (z)}}\left\lbrack {\left( {x - \frac{{\gamma (z)}x_{s}}{f}} \right)^{2} + \left( {y - \frac{{\gamma (z)}y_{s}}{f}} \right)^{2}} \right\rbrack} + {\; \theta}} \right\}} + {\exp \left\{ {{\frac{{- }\; \pi}{\lambda \; {\gamma (z)}}\left\lbrack {\left( {x - \frac{{\gamma (z)}x_{s}}{f}} \right)^{2} + \left( {y - \frac{{\gamma (z)}y_{s}}{f}} \right)^{2}} \right\rbrack} - {\; \theta}} \right\}}},} \right.}} & (50)\end{matrix}$

where, y(z)=[d₂−a−z(d₁a+d₂f−af+d₂a−d₁d₂)/f²]/[1−z(a+f−d₁)/f²]. For ageneral 3-D object g(x_(s),y_(s),z_(s)) illuminated by a narrowbandincoherent illumination, the intensity of the recorded hologram is anintegral of the entire PSFs given in Eq. (50), over all object pointsg(x_(s),y_(s),z_(s)), as follows

$\begin{matrix}{{H\left( {x,y} \right)} = {A{\quad\left( {C + {\int{\int{\int{{g\left( {x_{s},y_{s},z_{s}} \right)}\exp \left\{ {{\frac{\; \pi}{\lambda \; {\gamma (z)}}\left\lbrack {\left( {x - \frac{\gamma \; x_{s}}{f}} \right)^{2} + \left( {y - \frac{\gamma \; y_{s}}{f}} \right)^{2}} \right\rbrack} + {\; \theta}} \right\} {x_{s}}{y_{s}}{z_{s}}}}}} + {\int{\int{\int{{g\left( {x_{s},y_{s},z_{s}} \right)}\exp \left\{ {{\frac{{- }\; \pi}{\lambda \; {\gamma (z)}}\left\lbrack {\left( {x - \frac{\gamma \; x_{s}}{f}} \right)^{2} + \left( {y - \frac{\gamma \; y_{s}}{f}} \right)^{2}} \right\rbrack} - {\; \theta}} \right\} {x_{s}}{y_{s}}{{z_{s}}.}}}}}} \right.}}} & (51)\end{matrix}$

Besides a constant term Eq. (51) contain two terms of convolutionbetween an object and a quadratic phase, z-dependent, function, whichmeans that the recorded hologram is indeed a Fresnel hologram. In orderto remain with a single convolution term out of the three terms given inEq. (51), we follow the usual procedure of on-axis digital holography(I. Yamaguchi, and T. Zhang, “Phase-shifting digital holography,” Opt.Lett. 22, 1268-1269 (1997). commercial). Three holograms of the sameobject are recorded each of which with a different phase constant θ. Thefinal hologram H_(F) is a superposition according to the following,

H _(F)(x,y)=H ₁(x,y)[exp(−iθ ₃)−exp(−iθ ₂)]+H ₂(x,y)[exp(−iθ ₁)−exp(−iθ₃)]+H ₃(x,y)[exp(−iθ ₂)−exp(−iθ ₁)].  (52)

where H_(k) is the kth recorded hologram with the phase constant θ_(k).A 3D image s(x,y,z) can be reconstructed from H_(F)(x,y) by calculatingthe Fresnel propagation formula, as follows,

$\begin{matrix}{{{s\left( {x,y,z} \right)} = {{H_{F}\left( {x,y} \right)}*{\exp \left\lbrack {\frac{\; \pi}{\lambda \; z}\left( {x^{2} + y^{2}} \right)} \right\rbrack}}},} & (53)\end{matrix}$

where the asterisk denotes a 2D convolution.

The system shown in FIG. 33 has been used to record the three holograms.The SLM (HOLOEYE HEO 1080P) is phase-only and as so the desired functiongiven by Eq. (48) cannot be directly displayed on this SLM 3306. Toovercome this obstacle, we chose to display the phase function Q(−1/a)on only half of the SLM pixels. The rest of the pixels were modulatedwith a constant phase, where the pixels of each kind were selectedrandomly. By this method the SLM function becomes a good approximationto R(x,y) of Eq. (48). The SLM 3306 has 1920×1080 pixels in a display of16.6×10.2 mm, where only 1024×1024 pixels were used for implementing theDOE. The phase distribution of the three reflection masks displayed onthe SLM 3306, with phase constants of 0°, 120° and 240°, are shown inFIGS. 34A-34C, respectively. The other specifications of the system are:f=250 mm, a=800 mm, d₁=132 mm, d₂=260 mm. The system also includesbeamsplitter BS 3307 and lens L 3305 is spherical with a focal length f.

Three white on black letters each (U, S, and A), the 3D object 3302 ofthe size 2×2 mm were located at the vicinity of rear focal point of thelens. ‘O’ was at z=−24 mm, ‘S’ was at z=−48 mm and ‘A’ was at z=−72 mm.These letters were illuminated by a mercury arc lamp (Zeiss-AttoArc 2,HBO 100W). A filter which passed 574 to 725 nm light with a peakwavelength of 599 nm and a bandwidth of 60 nm was positioned between thebeamsplitter and the lens L. The three holograms, each for a differentphase constant of the DOE, were recorded by a cooled CCD camera(HAMAMATSU DIGITAL CAMERA C4742-95) and processed by a computer. Thefinal hologram H_(F)(x,y) was calculated according to Eq. (52) and itsmagnitude and phase distribution are depicted in FIGS. 34E and 34G,respectively.

The hologram H_(F)(x,y) was reconstructed in the computer by calculatingthe Fresnel propagation toward various propagation distances accordingto Eq. (53). Three different reconstruction planes are shown in FIGS.34G, 34H, and 34I. In each plane a different letter is in focus as isindeed expected from a holographic reconstruction of an object with avolume.

A process according to this invention may record holograms of realistic3-D objects illuminated by incoherent light. Since an embodiment of thesystem may have only a single channel, it is not affected by vibrations,it does not demand complicated alignment and the bandwidth can be widerthan conventional incoherent interferometers. The concept of the presentsystem can be applied to the design for a portable and very simpleholographic camera which might be useful for various applications in thefields of microscopy, still and video photography, astronomy and medicalimaging.

By this method, light is reflected from a 3-D object, propagates throughor is reflected from a diffractive optical element (DOE) and is recordedby a digital camera. Each beam which originates from any object point issplit into two different, mutually coherent, spherical waves. The beamsplitting is done by the DOE grating, which operates as if it were acomposition of two different diffractive spherical lenses. Therefore,the single wave-front originated from a point-source is divided by theDOE to two wave-fronts with different quadratic curves that propagate inthe same direction. The intensity of the two wave-front interference,originated from the same point source, is accumulated incoherently onthe camera pixel array with the other interferences from the entireobject points to yield the complete hologram. In order to get rid of thetwin image and the bias beam resulting from each single hologram whichwill be described later, three incoherent holograms are recordedsequentially, each with a different phase factor of the DOE. Using thecommon routines of digital holography (J. Rosen, G. Indebetouw, G.Brooker, Opt. Exp. 14, 4280-4285 (2006)) (J. Rosen, and G. Brooker“Incoherent digital holography,” Submitted for publication in Opt. Lett.(November 2006)) (I. Yamaguchi, and T. Zhang, Opt. Lett. 22, 1268-1269(1997)), the three holograms are superposed in the computer, such thatthe result is a complex valued Fresnel hologram. When this hologram isreconstructed in the computer, a single 3-D image of the object appearsin the digital reconstruction space.

The technique described above may also be used for color fluorescenceimaging. An example of such an embodiment produces a color Fresnelhologram which reconstructs the 3-D object with its original fluorescentcolors. The 3D objects that are imaged, for example “dice”, may containa fluorescent light source, such as several spots of two fluorescentdyes each with different emission wavelengths. The 3D objects areilluminated by an arc lamp source with a bandpass filter to illuminatethe specimen with incoherent light of about 50 nm bandwidth and whichcan also excite fluorescence in each of the fluorescent dyes. Accordingto this example, several digital holograms are generated for each of thedifferent fluorescent colors on the dice and for the dice themselves.Each emission color is introduced into the recording system byrestricting the emission with a specific chromatic filter. For eachwavelength of the fluorescence emission and the reflectednon-fluorescent light image of the object, a different Fresnel number isapplied to the DOE's grating. For each wavelength, three holograms aresequentially recorded, each with a different phase factor of the DOE'sfunction, such that the overall number of captured holograms for Mcolors plus the complete reflected non-fluorescent image of the objectis 3·(M+1). Every three holograms of the same wavelength are superposedin a certain way such that the result is a complex valued Fresnelhologram of this wavelength. The digital reconstruction from eachhologram is added to the rest, yielding a complete color 3-D image ofthe original object. To the best of our knowledge, the demonstratedholograms are the first fluorescence holograms recorded without scanningand the first fluorescence multiwavelength emission color holograms everrecorded.

An incoherent blue light source 3501 with a bandwidth of 56 nmilluminates a 3-D object 3502 as is shown in FIG. 35. The object'sfluorescent emission light 3503 is introduced into the system afterpassing through one of the chromatic filters F₂ 3504. After passingthrough lens L₁ 3505, the beam is reflected from a spatial lightmodulator (SLM) 3506 toward a demagnification setup of two lenses L₂3507 and L₃ 3508, which projects the holographic pattern onto a CCDcamera 3509. To understand the operational principle, we analyzed thesystem by following its response to an input object of a singleinfinitesimal point. Knowing the system's point spread function (PSF),enables one to analyze the system operation for any general object.

A Fresnel hologram of a point object is obtained when the twointerfering beams are, for instance, plane and spherical beams.Therefore, we choose the DOE's reflection function R(x_(D),y_(D))displayed on the SLM to be of the form,

$\begin{matrix}{{{R\left( {x_{D},y_{D}} \right)} = {\frac{1}{2} + {\frac{1}{2}{\exp \left\lbrack {{{- \frac{\; \pi}{\lambda \; a}}\left( {x_{D}^{2} + y_{D}^{2}} \right)} - {\; \theta}} \right\rbrack}}}},} & (54)\end{matrix}$

Where λ is the central wavelength introduced to the system. The constantterm of ½ in Eq. (54) contributes the plane wave, and the quadraticphase term is the paraxial approximation of the spherical wave. Theangle θ is the phase shift needed in order to get rid of the twin imageand the bias term.

The reflection function of the DOE given by Eq. (54) implies that thesystem's outcome can be viewed as a sum of two overlap imaging systems.Finding the location of each image is a key concept for understandingthis holographic recorder. In one system, let's call it system A, theDOE is actually a converging diffractive lens with a focal length of a,whereas in the other system (system B) the DOE serves as a plane mirror.In system A, a point source located at a distance d_(s)=f₁ from the lensL₁ is imaged to an image point at a distance (a−d₂)(f₃/f₂)² beyond thecamera plane. In this last expression we use the well-known fact thatthe axial magnification of an ordinary imaging system is given by therelation M_(A)=M_(T) ²=(d_(o)/d_(s))², where M₁ is the transversemagnification, and d_(o) is the distance from the output aperture to theimage. For any point at (0,0,z_(s)) located a distance d_(s)=f₁−z_(s)from the lens L₁, assuming that z_(s)<<f₁, the distance d_(o) isapproximately,

$\begin{matrix}{{{d_{o}\left( z_{s} \right)} \cong {{\left( {a - d_{2}} \right)\left( \frac{f_{3}}{f_{2}} \right)^{2}} + {{\overset{\_}{M}}_{A}z_{s}}}} = {{{\left( {a - d_{2}} \right)\left( \frac{f_{3}}{f_{2}} \right)^{2}} + {\left( \frac{f_{3}a}{f_{2}f_{1}} \right)^{2}z_{s}}} = {\left\lbrack {{\left( \frac{r^{2}}{f_{1}\lambda \; N} \right)^{2}z_{s}} + \frac{r^{2}}{\lambda \; N} - d_{2}} \right\rbrack \left( \frac{f_{3}}{f_{2}} \right)^{2}}}} & (55)\end{matrix}$

where the overall axial magnification M_(A) is the product ofmagnifications of the two consecutive imaging systems, r is the DOE'sradius and N is the DOE's Fresnel number given by N=r²/λa.

In system B, assuming that d_(s) f, the object point is obtained farbeyond the camera plane at a distance that justifies approximating thelocation of the image point at infinity. Therefore, for a point at(x_(s),y_(s),z_(s)), the intensity on the camera plane is the squaremagnitude of the complex amplitude sum of the spherical wave convergingat the distance d_(o) beyond the CCD plane, together with a plane wave,as follows,

$\begin{matrix}{{I_{P}\left( {x,y} \right)} \cong {C{{1 + {\exp \left\{ {{\frac{{- }\; \pi}{\lambda \; {d_{o}\left( z_{s} \right)}}\left\lbrack {\left( {x - {{\overset{\_}{M}}_{T}x_{s}}} \right)^{2} + \left( {y - {{\overset{\_}{M}}_{T}y_{s}}} \right)^{2}} \right\rbrack} - {\; \theta}} \right\}}}}^{2}}} & (56)\end{matrix}$

where the overall transverse magnification is M_(T)=f₃a/f₂f₁=f₃r²/λNf₂f₁. For a general 3-D objectg(x_(s),y_(s),z_(s)), illuminated by a narrowband incoherentillumination, the intensity of the recorded hologram is an integral overthe entire PSFs, given by Eq. (56), over all the object points, asfollows

$\begin{matrix}{{H\left( {x,y} \right)} = {{A\left( {C^{\prime} + {\int{\int{\int{{g\left( {x_{s},y_{s},z_{s}} \right)}\exp \left\{ {{\frac{\; \pi}{\lambda \; {d_{o}\left( z_{s} \right)}}\left\lbrack {\left( {x - {{\overset{\_}{M}}_{T}x_{s}}} \right)^{2} + \left( {y - {{\overset{\_}{M}}_{T}y_{s}}} \right)^{2}} \right\rbrack} + {\; \theta}} \right\} {x_{s}}{y_{s}}{z_{s}}}}}} + {\int{\int{\int{{g\left( {x_{s},y_{s},z_{s}} \right)}\exp \left\{ {{\frac{{- }\; \pi}{\lambda \; {d_{o}\left( z_{s} \right)}}\left\lbrack {\left( {x - {{\overset{\_}{M}}_{T}x_{s}}} \right)^{2} + \left( {y - {{\overset{\_}{M}}_{T}y_{s}}} \right)^{2}} \right\rbrack} - {\; \theta}} \right\} {x_{s}}{y_{s}}{z_{s}}}}}}} \right)}.}} & (57)\end{matrix}$

Besides a constant term C′, Eq. (57) contains two terms of correlationbetween an object and a quadratic phase, z_(s)-dependent, function,which means that the recorded hologram is indeed a Fresnel hologram. Inorder to remain with a single correlation term out of the three termsgiven in Eq. (57), we follow the procedure of on-axis digital holography(J. Rosen, G. Indebetouw, G. Brooker, Opt. Exp. 14, 4280-4285 (2006))(J. Rosen, and G. Brooker “Incoherent digital holography,” Submitted forpublication in Opt. Lett. (November 2006)). Three holograms of the sameobject are recorded each of which with a different phase constant θ. Thefinal hologram H_(F) is a superposition according to the following,

$\begin{matrix}{{H_{F}\left( {x,y} \right)} = {{{{H_{1}\left( {x,y} \right)}\left\lbrack {{\exp \left( {{- }\; \theta_{3}} \right)} - {\exp \left( {{- }\; \theta_{2}} \right)}} \right\rbrack} + {{H_{2}\left( {x,y} \right)}\left\lbrack {{\exp \left( {{- }\; \theta_{1}} \right)} - {\exp \left( {{- }\; \theta_{3}} \right)}} \right\rbrack} + {{{H_{3}\left( {x,y} \right)}\left\lbrack {{\exp \left( {{- }\; \theta_{2}} \right)} - {\exp \left( {{- }\; \theta_{1}} \right)}} \right\rbrack}.}} = {\int{\int{\int{{g\left( {x_{s},y_{s},z_{s}} \right)}\exp \left\{ {\frac{\; \pi}{\lambda \; {d_{o}\left( z_{s} \right)}}\left\lbrack {\left( {x - {{\overset{\_}{M}}_{T}x_{s}}} \right)^{2} + \left( {y - {{\overset{\_}{M}}_{T}x_{s}}} \right)^{2}} \right\rbrack} \right\} {x_{s}}{y_{s}}{{z_{s}}.}}}}}}} & (58)\end{matrix}$

where H_(k) is the k-th recorded hologram with the phase constant θ_(k)and k=1,2,3.

A 3-D image can be digitally reconstructed from H_(F)(x,y) bycalculating the Fresnel propagation (J. Goodman, Introduction to FourierOptics, 2^(nd) ed., McGraw-Hill, New York, 1996, pp. 63-95 (Chapter 4)).The reconstruction results of different chromatic holograms are composedtogether to a complete color figure. In order to get the same transverseand axial magnifications for all the wavelengths we change the Fresnelnumber of the DOE such that d_(o)(z_(s)), given by Eq. (55), remains thesame for all recorded wavelengths. In other words, the Fresnel number ofthe (i+1)-th wavelength λ_(i+1) is N_(i+1)=N_(i)λ_(i)/λ_(i+1), where N,is the Fresnel number of the i-th wavelength λ_(i).

An experiment showing the recording of a color fluorescence hologram wascarried out on the system shown in FIG. 35. The SLM (HOLOEYE HEO 1080P)is phase-only, and as so, the desired function given by Eq. (54) cannotbe directly displayed on this SLM. Instead, as a good approximation forEq. (54), we chose to display the required quadratic phase function ononly half of the SLM pixels. The rest of the pixels were modulated witha constant phase, where the pixels of both types were selected randomly(J. Rosen, and G. Brooker “Incoherent digital holography,” Submitted forpublication in Opt. Lett. (November 2006)). The central 1024×1024 pixelsof the SLM, on an area of 9.7 mm×9.7 mm, were used for displaying theDOE. The phase constants of θ_(1,2,3)=0°,120°,240° were introduced intothe three quadratic phase functions. The other specifications of thesystem are: f₁=250 mm, f₂=150 mm, f₃=35 mm, d₁=135 mm, d₂=206 mm. L₁,L₂, and L₃ are spherical lenses, and F₁ and F₂ are chromatic filters.

A pair of 8 mm×8 mm dice (i.e., the 3D object 3502) (in which some ofthe dots were painted with either red or green fluorescent paint) werepositioned at the vicinity of the rear focal point of lens L₁ 3505. Thecenter of the die with red fluorescent spots and the die with greenfluorescent spots were at a distance of 228 mm and 260 mm from L₁ 3505,respectevely. These dice were illuminated with a mercury arc lamp(ZEISS-ATTOARC 2, HBO 100W) in which only light from 444 to 500 nm witha peak wavelength of 472 nm and a bandwidth of 56 nm was allowed to passthrough bandpass filter F₁ 3510. All of the holograms were recorded by acooled CCD camera (HAMAMATSU DIGITAL CAMERA C4742-95, 12 bit, 1024×1280pixels, bin 1) and processed by a computer. The first three holograms(0, 120 and 240 degrees) of the non-fluorescent surfaces on the dicewere recorded with an identical filter as the source's filter mentionedabove placed in the emission filter slider F₂ 3504 The Fresnel numberfor these holograms was chosen to be N_(B)=10 (based upon a centerwavelength of 472 nm). The magnitude and phase of the final complexhologram, superposed from the first three holograms, is shown in FIGS.36( a) and (b), respectively. The reconstruction from the final hologramwas calculated using the Fresnel propagation formula. (J. Goodman,Introduction to Fourier Optics, 2^(nd) ed., McGraw-Hill, New York, 1996,pp. 63-95 (Chapter 4)) The results are shown at the plane of the frontface of the front die [36(c)], and at the plane of the front face of therear die [36(d)]. Note that in each plane a different die face is infocus as is indeed expected from a holographic reconstruction of anobject with a volume. The second set of three holograms was recorded viaa red filter in the emission filter slider F₂ 3504 which passed 614 to640 nm fluorescent light with a peak wavelength of 626 nm and abandwidth of 1 nm. The Fresnel number during the recording of the ‘red’holograms was N_(R=7.8). The magnitude and phase of the final complexhologram, superposed from the ‘red’ set, is shown in FIGS. 36( e) and(f), respectively. The reconstruction results from this final hologramare shown in FIGS. 36 (g) and (h) at the same planes as in FIGS. 36 (c)and (d), respectively. Finally, an additional set of three holograms wasrecorded with a green filter in emission filter slider F₂ 3504 whichpassed 500 to 532 nm fluorescent light with a peak wavelength of 516 nmand a bandwidth of 16 nm. The Fresnel number during the recording of the‘green’ holograms was N_(G)=9.2. The magnitude and phase of the finalcomplex hologram, superposed from the ‘green’ set, is shown in FIGS. 36(i) and (j), respectively. The reconstruction results from this finalhologram are shown in FIGS. 36 (k) and (l) at the same planes as inFIGS. 36 (c) and (d), respectively. Compositions of FIGS. 36( c), (g)and (k) and FIGS. 36( d), (h) and (l) are depicted in FIGS. 36( m) and(n), respectively. Note that all the colors in FIG. 36 arepseudo-colors. These last results yield a complete color 3-D holographicimage of the object including the red and green fluorescence. While theoptical arrangement in this demonstration has not been optimized formaximum resolution, it is important to recognize that even with thissimple optical arrangement, the resolution is good enough to image thefluorescent emissions with good fidelity and to obtain good reflectedlight images of the dice. Furthermore, in the reflected light images inFIGS. 36( c) and 36(d) the system has been able to detect reflections ofthe illumination on the dice.

Thus, this example shows a process of recording color holograms of 3-Dfluorescent objects. This example motionless system is not affected byvibrations, it does not require complicated alignment or a laser and thebandwidth can be wider than conventional incoherent interferometers,entirely because this holographic recorder is implemented on a singlechannel setup. The proposed design might play an important role in manytypes of 3D fluorescence applications (patents pending) includingfluorescence microscopy so that multicolor 3-D structures and dynamicprocesses could be imaged without any scanning, and therefore would beexpected to be faster then other methods.

Numerous modifications and variations of the present invention arepossible in light of the above teachings. It is therefore to beunderstood that within the scope of the appended claims, the inventionmay be practiced otherwise than as specifically described herein.

1. An apparatus configured to produce a hologram of an object, saidapparatus comprising: an electromagnetic radiation assembly consistingof one diffractive electromagnetic radiation element and configured toreceive a received electromagnetic radiation from the object andtransmit a transmitted electromagnetic radiation based on the receivedelectromagnetic radiation; and an image capture assembly configured tocapture an image of the transmitted electromagnetic radiation, andproduce the hologram of the object from the captured image, wherein theelectromagnetic radiation assembly is configured to control only a phaseof the transmitted electromagnetic radiation.